Department of

# Mathematics

Seminar Calendar
for events the year of Tuesday, January 1, 2019.

.
events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, January 14, 2019

3:00 pm in 343 Altgeld Hall,Monday, January 14, 2019

#### Organizational Meeting

###### Brian Shin (UIUC Math)

Tuesday, January 15, 2019

2:00 pm in 243 Altgeld Hall,Tuesday, January 15, 2019

#### Cut-edges and Regular Subgraphs in Odd-degree Regular Graphs

###### Douglas B. West (Zhejiang Normal University and University of Illinois)

Abstract: Hanson, Loten, and Toft proved that every $(2r+1)$-regular graph with at most $2r$ cut-edges has a $2$-factor. We generalize this by proving for $k\le(2r+1)/3$ that every $(2r+1)$-regular graph with at most $2r-3(k-1)$ cut-edges has a $2k$-factor. The restrictions on $k$ and on the number of cut-edges are sharp. We characterize the graphs with exactly $2r-3(k-1)+1$ cut-edges but no $2k$-factor. For $k>(2r+1)/3$, there are graphs without cut-edges that have no $2k$-factor. (Joint work with Alexandr V. Kostochka, Andr\'e Raspaud, Bjarne Toft, and Dara Zirlin.)

We determine the maximum guaranteed size of a $2$-regular subgraph in a $3$-regular $n$-vertex graph. In particular, we prove that every multigraph with maximum degree $3$ and exactly $c$ cut-edges has a $2$-regular subgraph that omits at most $(3n-2m+c-1)/2$ vertices (or $0$ for $3$-regular graphs without cut-edges). The bound is sharp; we describe the extremal multigraphs. (Joint work with Ilkyoo Choi, Ringi Kim, Alexandr V. Kostochka, and Boram Park.)

Wednesday, January 16, 2019

3:00 pm in 2 Illini Hall,Wednesday, January 16, 2019

#### Organizational Meeting

###### Sungwoo Nam (UIUC Math)

Thursday, January 17, 2019

11:00 am in 241 Altgeld Hall,Thursday, January 17, 2019

#### What is Carmichael's totient conjecture?

###### Kevin Ford (Illinois Math)

Abstract: A recent DriveTime commercial features a mathematician at a blackboard supposedly solving "Carmichael's totient conjecture". This is a real problem concerning Euler's $\phi$-function, and remains unsolved, despite the claim made in the ad. We will describe the history of the conjecture and what has been done to try to solve it.

12:00 pm in 243 Altgeld Hall,Thursday, January 17, 2019

#### Index properties of random automorphisms of free groups.

###### Ilya Kapovich (Hunter College)

Abstract: For automorphisms of the free group $F_r$, being "fully irreducible" is the main analog of the property of being a pseudo-Anosov element of the mapping class group. It has been known, because of general results about random walks on groups acting on Gromov-hyperbolic spaces, that a "random" (in the sense of being generated by a long random walk) element $\phi$ of $Out(F_r)$ is fully irreducible and atoroidal. But finer structural properties of such random fully irreducibles $\phi\in Out(F_r)$ have not been understood. We prove that for a "random" $\phi\in Out(F_r)$ (where $r\ge 3$), the attracting and repelling $\mathbb R$-trees of $\phi$ are trivalent, that is all of their branch points have valency three, and that these trees are non-geometric (and thus have index $<2r-2$). The talk is based on a joint paper with Joseph Maher, Samuel Taylor and Catherine Pfaff.

Friday, January 18, 2019

11:00 am in 145 Altgeld Hall,Friday, January 18, 2019

#### Organizational Meeting

###### Derek Kielty (Illinois Math)

4:00 pm in 141 Altgeld Hall,Friday, January 18, 2019

#### Organizational Meeting

Abstract: We will draft a schedule of the seminar talks this semester. Please join us and sign up if you want to speak (you don't have to decide on a topic or abstract now). As usual, there will be cookies. All are welcome!

4:00 pm in 345 Altgeld Hall,Friday, January 18, 2019

#### Generic flat pregeometries

###### Omer Mermelstein (University of Wisconsin, Madison.)

Abstract: The property of "flatness" of a pregeometry (matroid) is best known in model theory as the device with which Hrushovski showed that his example refuting Zilber's conjecture does not interpret an infinite group. I will dedicate the first part of this talk to explaining what flatness is, how it should be thought of, and how closely it relates to hypergraphs and Hrushovski's construction method. In the second part, I will conjecture that the family of flat pregeometries associated to strongly minimal sets is model theoretically nice, and share some intermediate results.

Tuesday, January 22, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, January 22, 2019

#### Singular limits of sign-changing weighted eigenproblems

###### Derek Kielty   [email] (Illinois Math)

Abstract: Eigenvalue problems with positive weights are related to heat flow and wave propagation in inhomogeneous media. Sign-changing weights have ecological interpretations, and generate spectra that accumulate at both positive and negative infinity. This talk will discuss recent results on limits of such eigenvalue problems when a negative portion of the weight is made arbitrarily large.

1:00 pm in Altgeld Hall,Tuesday, January 22, 2019

#### A Homotopical View of Lascar Groups of First-Order Theories

###### Greg Cousins (Notre Dame)

Abstract: In this talk, we will discuss how the Lascar group of a first-order theory, $T$, can be recovered as the fundamental group(-oid) of a certain space associated to the category of models, $Mod(T)$. We will then discuss some examples illustrating how tools from algebraic topology can be used to compute the Lascar group of a theory. Time permitting, we will discuss generalizations to the context of AECs and questions their higher homotopy. No knowledge of homotopy theory will be assumed. This is joint work with Tim Campion and Jinhe Ye.

2:00 pm in 243 Altgeld Hall,Tuesday, January 22, 2019

#### Ordered and convex geometric trees with linear extremal function

###### Alexandr Kostochka (Illinois Math)

Abstract: The extremal functions $\text{ex}_{\rightarrow}(n,F)$ and $\text{ex}_{\circ}(n,F)$ for ordered and convex geometric acyclic graphs $F$ have been extensively investigated by a number of researchers. Basic questions are to determine when $\text{ex}_{\rightarrow}(n,F)$ and $\text{ex}_{\circ}(n,F)$ are linear in $n$, the latter posed by Brass-Károlyi-Valtr in 2003. In this talk, we answer both these questions for every tree $F$.

We give a forbidden subgraph characterization for a family $\mathcal{ T}$ of ordered trees with $k$ edges, and show that $\text{ex}_{\rightarrow}(n,T) = (k - 1)n - {k \choose 2}$ for all $n \geq k + 1$ when $T \in {\mathcal T}$ and $\text{ex}_{\rightarrow}(n,T) = \Omega(n\log n)$ for $T \not\in {\mathcal T}$. We also describe the family ${\mathcal T}'$ of the convex geometric trees with linear Turán number and show that for every convex geometric tree $F\notin {\mathcal T}'$, $\text{ex}_{\circ}(n,F)= \Omega(n\log \log n)$.

This is joint work with Zoltan Füredi, Tao Jiang, Dhruv Mubayi and Jacques Verstraëte.

4:00 pm in 243 Altgeld Hall,Tuesday, January 22, 2019

#### Organizational Meeting

###### George Francis (University of Illinois/Urbana)

Abstract: Kay Kirkpatrick and George Francis invite you to join this seminar on machine learning (ML). It will be more of a mathematical learning collective than a show-and-tell venue. It meets in 243AH on Tuesdays at 4pm except when departmental events (colloquia, MSS and named lectures, spring departmental meeting) are held. Faculty, students, staff, and visitors are welcome. Our goal is to read and ponder papers, and ask each other many more questions than we expect to answer. For this organizational meeting we plan to collect topics you are interested in, and start a list of papers that might containthe answers. Please bring references to papers or websites you would like to study, either actively or passively. This way we might be able to come up with a tentative schedule of events.

Wednesday, January 23, 2019

3:00 pm in 2 Illini Hall,Wednesday, January 23, 2019

#### Torelli Theorem for curves

###### Lutian Zhao   [email] (UIUC Math)

Abstract: Jacobians are parametrizing the degree 0 line bundles. By sending a curve to its Jacobian we can get a polarized Abelian variety. The Torelli Theorem states we can reverse this map, i.e. for a polarized Abelian variety we can reconstruct the same curve. In this talk, I’ll start from Jacobian and prove the theorem. If time permitted, I’ll define the Torelli map for nodal curves.

Thursday, January 24, 2019

11:00 am in 241 Altgeld Hall,Thursday, January 24, 2019

#### Some statistics of the Euler phi function

###### Harold Diamond (Illinois Math)

Abstract: Questions about the distribution of value of the Euler phi function date to work of Schoenberg and Erdos. This talk will survey this theme and include a result of mine in which two applications of the Perron inversion formula are applied to count the number of points (n, phi(n)) lying in a specified rectangle.

12:00 pm in 243 Altgeld Hall,Thursday, January 24, 2019

#### Stretch Factors Coming From Thurston's Construction

###### Joshua Pankau (U Iowa)

Abstract: Associated to every pseudo-Anosov map is a real number called its stretch factor. Thurston proved that stretch factors are algebraic units, but it is unknown exactly which algebraic units are stretch factors. In this talk I will discuss a construction of pseudo-Anosov maps due to Thurston, and discuss my recent results where I classified (up to power) the stretch factors coming from this construction. We will primarily focus on a specific class of algebraic units known as Salem numbers. This talk is intended to be accessible to everyone.

1:00 pm in 347 Altgeld Hall,Thursday, January 24, 2019

#### Constraining neural networks with spiking statistics

###### Andrea Barreiro (Mathematics, Southern Methodist University)

Abstract: As experimental tools in neuroscience have advanced, measuring whole-brain dynamics with single-neuron resolution is becoming closer to reality. However, a task that remains technically elusive is to measure the interactions within and across brain regions that govern such system-wide dynamics. We propose a method to derive constraints on hard-to-measure neural network attributes --- such as inter-region synaptic strengths --- using easy-to-measure spiking statistics. As a test case, we studied interactions in the olfactory system. We used two micro-electrode arrays to simultaneously record from olfactory bulb (OB) and anterior piriform cortex (PC) of anesthetized rats who were exposed to several odors. We were able to make several predictions about the network, notably that inhibition within the OB and inhibition within PC were constrained to a narrow slice of possible values. As time permits, I’ll describe ongoing work in which we are applying the same techniques to determine how peripheral sensation and lateral inhibition combine to shape dynamic selectivity within the OB.

2:00 pm in 347 Altgeld Hall,Thursday, January 24, 2019

#### Introduction to Percolation Theory

###### Grigory Terlov (UIUC Math)

Abstract: This is the first part of two talks designed to introduce students to Percolation Theory. We will describe the model, talk about infinite clusters, prove the existence of the phase transition, introduce the universality principle and more.

Friday, January 25, 2019

2:00 pm in 141 Altgeld,Friday, January 25, 2019

#### A potential theoretic approach to box counting and packing dimensions

###### Fernando Roman-Garcia (Illinois Math)

Abstract: In 1968 Robert Kaufman introduced a potential theoretic approach to Hausdorff dimension. This approach allowed the use of Fourier analytic tools to answer questions about fractal Hausdorff dimension. In the late 90's Kenneth Falconer introduced a similar approach to packing and box counting dimensions. This allowed further developments on this area of geometric analysis such as Marstrand-Mattila type projection theorems for these different notions of fractal dimension. In this talk we will go through the development of this approach and (if time permits) go over the proof of the projection theorem for box and packing dimensions.

4:00 pm in 141 Altgeld Hall,Friday, January 25, 2019

#### Symmetric functions and Hilbert schemes

###### Joshua Wen (UIUC)

Abstract: One source of applications of geometric and topological methods to combinatorics and representation theory is to proving various numbers are positive integers by showing that said numbers are dimensions of some vector space. A big example of this from more than a decade ago is Haiman’s proof of the Macdonald positivity conjecture, which further cemented an already tight connection between symmetric functions and the topology of Hilbert schemes of points in $\mathbb{C}^2$. I want to go through this story while highlighting two lessons that nobody taught me in grad school—that generating series are awesome for geometers and how to do geometry on a moduli space.

Saturday, January 26, 2019

1:00 pm in Altgeld Hall,Saturday, January 26, 2019

#### To Be Announced

Monday, January 28, 2019

3:00 pm in 243 Altgeld Hall,Monday, January 28, 2019

#### Circle actions on almost complex manifolds with few fixed points

###### Donghoon Jang (Pusan National University)

Abstract: A circle action on a manifold can be thought of as a periodic flow on a manifold (periodic dynamical system), or roughly a rotation of a manifold. During this talk, we consider circle actions on almost complex manifolds, which are more general than symplectic manifolds. We discuss classification of a circle action on a compact almost complex manifold $M$, when the number $k$ of fixed points is small. If $k=1$, $M$ is a point. If $k=2$, $M$ resembles $S^2$ or $S^6$. If $k=3$, $M$ resembles $\mathbb{CP}^2$. We also discuss when $k=4$ and $\dim M \leq 6$. Techniques include equivariant cohomology and index theory.

5:00 pm in 241 Altgeld Hall,Monday, January 28, 2019

#### Quantum Null Energy Conjecture

###### Tom Faukner (UIUC physics)

Abstract: We will explain and discuss a recent result on the Quantum Null Energy Conjecture

Tuesday, January 29, 2019

12:00 pm in 243 Altgeld Hall,Tuesday, January 29, 2019

#### The Farey Sequence Next-Term Algorithm, and the Boca-Cobeli-Zaharescu Map Analogue for Hecke Triangle Groups G_q

###### Diaaeldin Taha (University of Washington)

Abstract: The Farey sequence is a famous enumeration of the rationals that permeates number theory. In the early 2000s, F. Boca, C. Cobeli, and A. Zaharescu encoded a surprisingly simple algorithm for generating--in increasing order--the elements of each level of the Farey sequence as what grew to be known as the BCZ map, and demonstrated how that map can be used to study the statistics of subsets of the Farey fractions. In this talk, we present a generalization of the BCZ map to all Hekce triangle groups G_q, q \geq 3, with the G_3 = SL(2, \mathbb{Z}) case being the "classical" BCZ map. If time permits, we will present some applications of the G_q-BCZ maps to the statistics of the discrete G_q linear orbits in the plane \mathbb{R}^2 (i.e. the discrete sets \Lambda_q = G_q (1, 0)^T).

1:00 pm in 347 Altgeld Hall,Tuesday, January 29, 2019

#### Traveling waves in an inclined channel and their stability

###### Zhao Yang (Indiana University Bloomington)

Abstract: The inviscid Saint-Venant equations are commonly used to model fluid flow in a dam or spillway. To classify known traveling wave solutions to the St. Venant equations, the condition of hydrodynamic stability introduces a dichotomy on the parameter F (Froude number): Namely, the constant flow solution is stable for F < 2 where one expect persistent asymptotically-constant traveling wave solutions and unstable for F > 2 where one expect rather complex pattern formation. We will discuss for F>2 Dressler's construction of the inviscid roll wave solution and for F<2 Yang-Zumbrun's construction of the smooth/discontinuous hydraulic shock profiles. We will then present recent stability results of these traveling waves. That is a complete spectral stability diagram for F>2 roll wave case obtained in [JNRYZ18] and spectral, linear orbital, and nonlinear orbital stability of all the hydraulic shock profiles obtained in [YZ18] and [SYZ18].

1:00 pm in 345 Altgeld Hall,Tuesday, January 29, 2019

#### Self-similar structures

###### Garret Ervin (Carnegie Mellon)

Abstract: An iterated function system is a finite collection $f_1, …, f_n$ of contraction mappings on a complete metric space. Every such system determines a unique compact subspace $X$, called the attractor of the system, such that $X = \bigcup f_i[X]$. Many well-known fractals, like the Cantor set and Sierpinski triangle, are realized as attractors of iterated function systems.
A surprisingly rich analysis can be carried out even when the functions $f_i$ are only assumed to be non-surjective injections from a set to itself. Moreover, in many cases this analysis can be used to characterize when a structure $X$, like a group or linear order, is isomorphic to a product of itself, or to its own square. Such structures behave much like attractors of iterated function systems. We present the technique, and cite solutions to two old problems of Sierpinski as an application.

2:00 pm in 243 Altgeld Hall,Tuesday, January 29, 2019

#### Eigenvalues and graph factors

###### Suil O (Stony Brook University)

Abstract: An (even or odd) $[a,b]$-factor is a spanning subgraph $H$ such that ($d_H(v)$ is even or odd respectively, and) $a \le d_H(v) \le b$ for all $v \in V(G)$. When $a=b=k$, it is called a $k$-factor.

In this talk, we give sharp conditions for a graph to have an even $[a,b]$-factor. For a positive integer $k$, we also prove a sharp lower bound for the spectral radius in an $n$-vertex graph to have a $k$-factor. Furthermore, we give a sharp lower bound for the third largest eigenvalue in an $n$-vertex $r$-regular graph to have odd $[1,b]$-factor.

This is joint work partly with Eun-Kyung Cho, Jongyoon Hyun, Jeongrae Park, and Douglas B. West.

3:00 pm in 243 Altgeld Hall,Tuesday, January 29, 2019

#### Non-reduced Parabolic Group Schemes

###### William Haboush (UIUC Math)

Abstract: In the 90’s I and my student N. Lauritzen described all possible non reduced parabolic subgroup schemes of a semisimple algebraic group. These lead to complete homogeneous spaces with very interesting properties. Among other things they provide counterexamples which were crucial to the Mori program. Now that the Lusztig conjecture has been shown to be completely false I am revisiting this material hoping to make some interesting contribution to the decomposition problem for Weyl modules.

4:00 pm in 243 Altgeld Hall,Tuesday, January 29, 2019

#### Backprop in Neural Nets and Automatic Differentiation

###### George Francis   [email] (University of Illinois at Urbana–Champaign)

Abstract: In 1988 Rumelhart et al brought backpropagation into prominence throughout the Connectionist School of AI (neural nets, hidden layers, deep learning, etc). The technique was used earlier, but had remained obscure til then. Now, 3 decades later, backprop is a well established component of ML theory and practice. But it often comes wrapped in dense mathematical obscurity. In my latter day efforts to understand backprop I finally found some comprehensible answers in Baydin, Pearlmutter, Radul, and Siskind's survey paper "Automatic Differentiation in Machine Learning", J. Machine Learning Res 18 (2018) pp 1-43. I hope to pass along what I learned by working through a very illuminating example, leaving the context and (informal) definitions to the ample Q/A part of the seminar. For more information about our seminar, please see its webpage at http://new.math.uiuc.edu/MathMLseminar/

Wednesday, January 30, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, January 30, 2019

#### The cube problem and Schroeder-Bernstein problem for linear orders

###### Garret Ervin (Carnegie Mellon)

Abstract: We sketch proofs of solutions to two old problems posed by Sierpinski concerning products of linear orders. The first problem asks whether there exists a linear order $X$ that is isomorphic to its lexicographic cube but not to its square; the second, whether there are two non-isomorphic orders $Y$ and $Z$ that divide each other on both the left and right side. For other classes of structures, the corresponding questions are usually either both positive or both negative, but for linear orders the answers diverge: there is no such $X$, but there are such $Y$ and $Z$.

3:00 pm in 2 Illini Hall,Wednesday, January 30, 2019

#### Canceled

4:00 pm in 245 Altgeld Hall,Wednesday, January 30, 2019

#### Rescheduled

Thursday, January 31, 2019

11:00 am in 241 Altgeld Hall,Thursday, January 31, 2019

#### Monodromy for some rank two Galois representations over CM fields

###### Patrick Allen (Illinois Math)

Abstract: In the automorphic-to-Galois direction of Langlands reciprocity, one aims to construct a Galois representation whose Frobenius eigenvalues are determined by the Hecke eigenvalues at unramified places. It is natural to ask what happens at the ramified places, a problem known a local-global compatibility. Varma proved that the p-adic Galois representations constructed by Harris-Lan-Taylor-Thorne satisfy local-global compatibility at all places away from p, up to the so-called monodromy operator. Using recently developed automorphy lifting theorems and a strategy of Luu, we prove the existence of the monodromy operator for some of these Galois representations in rank two. This is joint work with James Newton.

2:00 pm in 347 Altgeld Hall,Thursday, January 31, 2019

#### Introduction to Percolation Theory (Part 2)

###### Grigory Terlov (UIUC Math)

Abstract: This is the second part of two talks designed to introduce students to Percolation Theory. We will discuss an upper bound for critical probability for $\mathbb{Z}^d$ via cut-sets and duality. This talk should be accessible for people who missed the first part.

Friday, February 1, 2019

2:00 pm in 141 Altgeld Hall,Friday, February 1, 2019

#### A Heat Trace Anomaly on Polygons

Abstract: Given a planar domain with smooth boundary, one can associate its heat kernel, a time dependent operator whose trace admits an asymptotic expansion in t. The coefficients in this expansion turn out to all be geometric/topological invariants of the domain. However, by considering a smooth family of domains converging to a polygon, one can conclude that these heat trace coefficients are not continuous under such domain deformation. In this talk I’ll describe work of Mazzeo-Rowlett which recasts this apparent anomaly using renormalized invariants. I’ll also use it as an excuse to talk about uncommon but useful techniques in the study of linear PDEs e.g.: domain blow-ups, polyhomogeneous expansions, and more.

4:00 pm in 145 Altgeld Hall,Friday, February 1, 2019

#### Vector fields on Spheres

###### Brian Shin (UIUC)

Abstract: In this talk, I would like to tell the story of one of the classical problems in topology: how many pointwise linearly independent vector fields can you put on a sphere of dimension $n$. The famous Hairy Ball Theorem tells us that there are none if $n$ is even. On the other hand, if $n$ is one of 1, 3, or 7, we can construct $n$ such vector fields using the normed divison $\mathbb{R}$-algebra structures on complex numbers, quaternions, and octonions. In this talk, we'll discuss the complete resolution of this problem by Adams, using methods of geometry, algebra, and homotopy theory along the way.

4:00 pm in 245 Altgeld Hall,Friday, February 1, 2019

#### Nice rack: The evolution of deer antlers and other mating displays

###### Dr. Sara Clifton   [email] (UIUC Math)

Abstract: Species spanning the animal kingdom have evolved extravagant and costly ornaments to attract mating partners. Zahavi's handicap principle offers an elegant explanation for this: ornaments signal individual quality and must be costly to ensure honest signaling, making mate selection more efficient. Here, we incorporate the assumptions of the handicap principle into a mathematical model and show that they are sufficient to explain the heretofore puzzling observation of bimodally distributed ornament sizes in a variety of species.

Monday, February 4, 2019

12:00 pm in 343 Altgeld Hall,Monday, February 4, 2019

#### The integration problem for Courant algebroids

###### Rajan Mehta (Smith College)

Abstract: Courant algebroids originally appeared in the study of constrained Hamiltonian systems, but they are connected to many areas of mathematical physics, including multisymplectic geometry, double field theory, and (my personal interest) 3-dimensional topological field theory. Since a Courant structure involves a bracket that resembles a Lie bracket (but fails to be skew-symmetric), one might expect there to be some groupoid-like structure for which a Courant algebroid is the infinitesimal object. There is reason to believe that the answer should be a "symplectic 2-groupoid," but there are many devils in the details, including even the question of how "symplectic 2-groupoid" should be defined. I will describe various developments in this problem.

5:00 pm in 241 Altgeld Hall,Monday, February 4, 2019

#### A brief introduction to differential and Riemannian geometry

Tuesday, February 5, 2019

12:00 pm in 243 Altgeld Hall,Tuesday, February 5, 2019

#### Asymptotics of the expected diameter of translation surfaces

###### Anja Randecker (U Toronto)

Abstract: For the hyperbolic structure on a Riemann surface, Mirzakhani has proven asymptotics of the expected diameter for large genus surfaces. An abelian differential equips a Riemann surface with a translation structure. In joint work with Howard Masur and Kasra Rafi, we prove asymptotics for large genus translation surfaces of area 1. Unlike in the case of hyperbolic surfaces, the expected diameter goes to zero as the genus goes to infinity.

1:00 pm in 345 Altgeld Hall,Tuesday, February 5, 2019

#### Local Keisler Measures and NIP Formulas

###### Kyle Gannon (Notre Dame)

Abstract: The connection between finitely additive probability measures and NIP theories was first noticed by Keisler. Around 20 years later, the work of Hrushovski, Peterzil, Pillay, and Simon greatly expanded this connection. Out of this research came the concept of generically stable measures. In the context of NIP theories, these particular measures exhibit stable behavior. In particular, Hrushovski, Pillay, and Simon demonstrated that generically stable measures admit a natural finite approximation. In this talk, we will discuss generically stable measures in the local setting. We will describe connections between these measures and concepts in functional analysis as well as show that this interpretation allows us to derive a local approximation theorem.

2:00 pm in 243 Altgeld Hall,Tuesday, February 5, 2019

#### Fractalizers

###### Florian Pfender (University of Colorado Denver Math)

Abstract: A graph $H$ is a fractalizer if every graph $G$ maximizing the number of induced copies of $H$ is an iterated balanced blow-up of $H$. Fox, Hao and Lee, and independently Yuster, showed that almost every graph is a fractalizer considering random graphs. Nevertheless, no non-trivial explicit examples of fractalizers are known. We show that the cycle $C_5$ is almost a fractalizer, and conjecture that all longer cycles are fractalizers.

3:00 pm in 243 Altgeld Hall,Tuesday, February 5, 2019

#### Non-reduced Parabolic Group Schemes, II

###### William Haboush (UIUC Math)

4:05 pm in 243 AH,Tuesday, February 5, 2019

#### Backprop in NN and AD cont'd

###### George Francis   [email] (University of Illinois at Urbana)

Abstract: I will finish presenting some items in the handout last week. In particular I hope to explain just how Trask's updating the weights in his program for a machine to learn XOR might be derived from Pearlmutter&Siskind's reverse automatic differentiation recipe. This won't take the entire time, and I hope to answer questions and ask a few myself. There will no new items introduced and the seminar may end early. The temperature is forecasts to be 53F, but with showers.

Wednesday, February 6, 2019

3:00 pm in Altgeld Hall,Wednesday, February 6, 2019

#### Murphy's law in Hilbert scheme

###### Sungwoo Nam (Illinois Math)

Abstract: One feature of moduli space is that although it parametrizes nice objects like smooth projective curves, it can be quite bad. In this talk, we will see lots of instances of these phenomena(mostly involving lots of cohomology computations) focusing on Hilbert scheme of curves in a projective space. I'll end with a discussion on Mumford's famous pathological example and Murphy's law formulated by Vakil.

3:00 pm in 243 Altgeld Hall,Wednesday, February 6, 2019

#### Global climate change, regional climate impacts, and quantifying relevant uncertainties

###### Ryan Sriver (University of Illinois, Atmospheric Sciences)

Abstract: Earth is warming, and the damages associated with climate and weather extremes (droughts, heatwaves, hurricanes) are increasing. Projecting these changes into the future is difficult due to: incomplete understanding of the physical processes, inadequate numerical models and resolution, and relatively short observational records. Here we highlight some of the current grand climate change problems, and we present some of our group's recent work in areas related to climate extremes and uncertainty quantification surrounding projections of future climate change.

3:00 pm in 341 Altgeld Hall,Wednesday, February 6, 2019

#### "Generic representations of abelian groups and extreme amenability" by J. Melleray and T. Tsankov (Part 1)

###### Dakota Ihli (UIUC)

Abstract: In this series of talks, we discuss the paper in the title [arXiv link].

4:00 pm in 245 Altgeld Hall,Wednesday, February 6, 2019

#### Systems of Calogero-Moser Type

###### Matej Penciak (Illinois Math)

Abstract: It is well known that many-particle systems are in general not solvable analytically. For some specific choices of interactions between particles though, a lot can be said. In this talk I aim to give an introduction to systems of Calogero-Moser type and the surprising role of algebraic geometry in their solvability. I will also give a perspective on how this subject plays a role in some hot topics in mathematics in general: Hitchin integrable systems, geometric representation theory, and the geometric Langlands philosophy.

Thursday, February 7, 2019

2:00 pm in 347 Altgeld Hall,Thursday, February 7, 2019

#### An Introduction to Dyson Brownian Motion and Universality

###### Kesav Krishnan (UIUC Math)

Abstract: We define Brownian motion on the space of N×N Hermitian Matrices, and derive an SDE for the corresponding process of the eigenvalues. We then establish that the eigenvalue process is identical to Brownian motion in R^n confined to the Weyl Chamber.

Friday, February 8, 2019

4:00 pm in 245 Altgeld Hall,Friday, February 8, 2019

#### Can you be both central and vacant? A study of a small pond network

###### Prof. Zoi Rapti   [email] (UIUC Math)

Abstract: We will introduce a simple one-dimensional ordinary differential equation (Levin's equation) and concepts from network theory to analyze occupancy patterns in a small network of freshwater ponds. We will investigate various factors that determine whether a pond can be vacant or occupied by our organism (Daphnia pulex aka waterflea), which is prevalent in ponds and lakes of the Midwest. No knowledge of differential equations or network theory will be assumed: all background will be introduced in the talk.

4:00 pm in 345 Altgeld Hall,Friday, February 8, 2019

#### "The complexity of topological group isomorphism" by A. Kechris, A. Nies, and K. Tent (Part 1)

###### Anush Tserunyan (UIUC)

Abstract: This will be the introductory talk of the series on the paper in the title [arXiv link], which deals with the classification of some natural classes of non-Archimedean groups (= closed subgroups of S) up to topological group isomorphism. It gives a general criterion for a class of non-Archimedean groups to show that the topological group isomorphism on it is Borel-classifiable by countable structures. This criterion is satisfied by the classes of profinite groups, locally compact non-Archimedean groups, and oligomorphic groups.

4:00 pm in 145 Altgeld Hall,Friday, February 8, 2019

#### Hamiltonian Lie algebroids

###### Luka Zwaan (UIUC)

Abstract: Hamiltonian Lie algebroids were introduced quite recently by Blohmann and Weinstein, resulting from their work in general relativity. They are a generalisation of the usual notion of a Hamiltonian action of a Lie algebra on a presymplectic manifold to arbitrary Lie algebroids. In this talk, I will quickly recall this usual notion, and then discuss several ways Blohmann and Weinstein tried to generalise it. In the end, the most convenient method makes use of a choice of connection on the Lie algebroid.

Monday, February 11, 2019

3:00 pm in 243 Altgeld Hall,Monday, February 11, 2019

#### Rigidity of Lie groupoids and foliations

###### Rui Loja Fernandes (UIUC)

Abstract: I will discuss a result stating that a compact, Hausdorff, Lie groupoid is rigid. i.e., has no non-trivial deformations. As an application of this result, it follows that a compact, Hausdorff foliation is rigid if and only if the generic leaf has trivial 1st cohomology. This is closely related to old stability results for foliations due to Epstein, Rosenberg and Hamilton. This talk is based on joint work with Matias del Hoyo.

5:00 pm in 241 Altgeld Hall,Monday, February 11, 2019

#### Introduction to differential and Riemannian geometry part II

Tuesday, February 12, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, February 12, 2019

#### The Open Graph Dichotomy and the second level of the Borel hierarchy

###### Raphaël Carroy (Gödel Research Center for Math. Logic at Univ. of Vienna)

Abstract: I will explain how variants of the open graph dichotomy can be used to obtain various descriptive-set-theoretical dichotomies at the second level of the Borel hierarchy. This shows how to generalise these dichotomies from analytic metric spaces to separable metric spaces by working under the axiom of determinacy. If time allows it, I will also discuss some connections between cardinal invariants and the chromatic number of the graphs at stake.

2:00 pm in 243 Altgeld Hall,Tuesday, February 12, 2019

#### On the number of edges in C_5-free 3-uniform hypergraphs

###### Dara Zirlin (Illinois Math)

Abstract: In a 3-uniform hypergraph, a Berge 5-cycle is formed by five distinct edges $e_1,\dots e_5$ and five distinct vertices $v_1,\dots, v_5$, such that $v_i,v_{i+1}\in e_i$, where indices count modulo 5.

In 2007, Bollobás and Györi gave upper bounds on the number of triangles in a $C_5$-free graph, and on the number of edges in a 3-uniform hypergraph containing no Berge 5-cycles.
We improve their second bound. This is joint work with Alexandr Kostochka.

4:00 pm in Altgeld Hall,Tuesday, February 12, 2019

#### No Seminar Today

Abstract: To encourage faculty members of the seminar to join the 4pm departmental discussion in 245 of the math building design we won't have a seminar today. It will resume next week at the usual time and location.

4:00 pm in 245 Altgeld Hall,Tuesday, February 12, 2019

#### Altgeld-Illini Renovation/Building Project Feedback Session

Abstract: The department's Altgeld-Illini Renovation Committee seeks input from every member of the department to help us develop a clear vision of what we want in the new building and the renovations.

Wednesday, February 13, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, February 13, 2019

#### Introduction to well and even better quasi-orders

###### Raphaël Carroy (Gödel Research Center for Math. Logic, Univ. of Vienna)

Abstract: Well-quasi-orders, or wqos, generalize well-orders in the context of partial orders. They appear naturally in various domains of mathematics, and have been frequently rediscovered. I'll briefly explain why, and what we can do with them. I'll then talk about their limitations and why it's hard to prove that non-trivial quasi-orders are wqo. I will also show how trying to fix these problems leads to the definition of a smaller class of quasi-orders: better-quasi-orders, or bqos. If time allows, I'll get a bit into bqo theory.

3:00 pm in 243 Altgeld Hall,Wednesday, February 13, 2019

#### To Be Announced

###### Chen Chen (University of Chicago, Geophysical Sciences)

3:00 pm in 2 Illini Hall,Wednesday, February 13, 2019

#### Equivariant Cohomology

###### Ciaran O'Neill (Illinois Math)

Abstract: I’ll define equivariant cohomology and give some basic examples. Then I’ll go into more detail for the case of a torus action on projective space.

Thursday, February 14, 2019

12:00 pm in 243 Altgeld Hall,Thursday, February 14, 2019

#### Spectral Rigidity of q-differential Metrics

###### Marissa Loving (UIUC Math)

Abstract: When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask which curves’ lengths do we really need to know? It is a classical result of Fricke that a hyperbolic metric on a surface is determined by its marked simple length spectrum. More recently, Duchin–Leininger–Rafi proved that a flat metric induced by a unit-norm quadratic differential is also determined by its marked simple length spectrum. In this talk, I will describe a generalization of the notion of simple curves to that of q-simple curves, for any positive integer q, and show that the lengths of q-simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a q-differential metric.

2:00 pm in 347 Altgeld Hall,Thursday, February 14, 2019

#### An Introduction to Dyson Brownian Motion and Universality (Part 2)

###### Kesav Krishnan (UIUC Math)

Abstract: We will discuss the connections of Dyson Brownian Motion and the Totally Asymmetric Simple Exclusion Process (TASEP). This will be the first glimpse of the Kardar Parisi Zhang Universality class.

3:00 pm in 347 Altgeld Hall,Thursday, February 14, 2019

#### Quiver varieties and root multiplicities for symmetric Kac-Moody algebras

###### Peter Tingley   [email] (Loyola University, Chicago)

Abstract: We discuss combinatorial upper bounds on dimensions of certain imaginary root spaces for symmetric Kac-Moody algebras. These come from a realization of the infinity crystal using quiver varieties. The framework is quite general, but we only work out specifics for one special case. We conjecture that our bound is quite tight, and give both computational evidence and heuristic justification for this conjecture, but unfortunately not a proof.

Friday, February 15, 2019

2:00 pm in 141 Altgeld Hall,Friday, February 15, 2019

#### Convex geometry and the Mahler conjecture

###### Derek Kielty (Illinois Math)

Abstract: In this talk we will give an introduction to convex geometry and discuss the Mahler conjecture. This conjecture asserts that the product of the volume of a centrally symmetric convex set and the volume of its dual is minimized on a certain family of polytopes. We will also discuss a PDE analog of this conjecture.

3:00 pm in 341 Altgeld Hall ,Friday, February 15, 2019

#### Note the time and room change!"The complexity of topological group isomorphism" by A. Kechris, A. Nies, and K. Tent (Part 2)

###### Jenna Zomback (UIUC)

Abstract: This will be the second talk of the series on the paper in the title [arXiv link], which deals with the classification of some natural classes of non-Archimedean groups (= closed subgroups of S) up to topological group isomorphism. It gives a general criterion for a class of non-Archimedean groups to show that the topological group isomorphism on it is Borel-classifiable by countable structures. This criterion is satisfied by the classes of profinite groups, locally compact non-Archimedean groups, and oligomorphic groups. In this talk, we will fill in some proofs left out last time and prove this general criterion.

4:00 pm in Altgeld Hall 145 ,Friday, February 15, 2019

#### Laplacian Operator and Hyperbolic Geometry

###### Xiaolong Han (Illinois Math)

Abstract: The Laplacian operator acting on functions on a Riemannian manifold is an analytic operator invariant under isometry of the manifold. Its spectrum encodes much geometric information of the manifold. In this talk, I will start with some basic properties of Laplacian operator and hyperbolic geometry. Then I will talk about how these two interact with each other. Time permitting, I will talk about some of my recent works. No background on Laplacian operator or hyperbolic geometry is assumed.

4:00 pm in 241 Altgeld Hall,Friday, February 15, 2019

#### How to Give a Good Math Talk

###### uAWM, MATRIX, & IGL Outreach   [email] (UIUC Math)

Abstract: We will be having a workshop for undergraduates wishing to present and give a talk in the Undergraduate Seminar this semester (and in the future). We'll go through all the basics of giving an interesting talk as well as some details that can really make a presentation stand out.

4:00 pm in 245 Altgeld Hall,Friday, February 15, 2019

#### Harry Potter's Cloak Via Transformation Optics

###### Gunther Uhlmann (University of Washington)

Abstract: Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc. including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last fifteen years or so there have been several scientific proposals to achieve invisibility. We will introduce in a non-technical fashion one of them, the so-called "transformation optics" which has received a lot of attention in the scientific community.

Monday, February 18, 2019

1:00 pm in Altgeld Hall,Monday, February 18, 2019

#### To Be Announced

3:00 pm in 243 Altgeld Hall,Monday, February 18, 2019

#### Swindles relating distinct symplectic structures

###### James Pascaleff (UIUC)

Abstract: An interesting phenomenon in symplectic topology is the existence of multiple non-equivalent symplectic structures on a single manifold. Often, such structures can be distinguished by their Fukaya categories. A natural question is whether there is any relationship between these categories. In this talk I will show that in some simple examples the categories are related by functors that are reminiscent of the Eilenberg swindle.

3:00 pm in 341 Altgeld Hall,Monday, February 18, 2019

#### Dense orbits in the space of subequivalence relations

###### Forte Shinko (Caltech)

Abstract: Given a measure-preserving countable Borel equivalence relation $E$, there is a Polish space $S(E)$ of subequivalence relations, which admits a natural action of the full group $[E]$. One can ask the following natural question: does $S(E)$ have a dense orbit? We will present results due to François Le Maître which show that the answer is yes when $E$ is the hyperfinite ergodic equivalence relation, and that the answer is no when $E$ is induced by a measure-preserving action of a property (T) group.

4:00 pm in 245 Altgeld Hall,Monday, February 18, 2019

#### Cohomology of Shimura Varieties

###### Sug Woo Shin (University of California Berkeley)

Abstract: Shimura varieties are a certain class of algebraic varieties over number fields with lots of symmetries, introduced by Shimura and Deligne nearly half a century ago. They have been playing a central role in number theory and other areas. Langlands proposed a program to compute the L-functions and cohomology of Shimura varieites in 1970s; this was refined by Langlands-Rapoport and Kottwitz in 1980s. I will review some old and recent results in this direction.

5:00 pm in 241 Altgeld Hall,Monday, February 18, 2019

#### Introduction to differential and Riemannian geometry part III

Tuesday, February 19, 2019

11:00 am in 345 Altgeld Hall ,Tuesday, February 19, 2019

#### G-equivariant factorization algebras

###### Laura Wells (Notre Dame Math)

Abstract: Factorization algebras are a mathematical tool used to encode the data of the observables of a field theory. There are various notions of factorization algebra: one can define a factorization algebra on the open subsets of some fixed manifold; or alternatively, one can define a factorization algebra on the site of all manifolds of a given dimension with specified geometric structure. In this talk I will outline a comparison between two such notions: G-equivariant factorization algebras on a fixed model space M and factorization algebras on the site of all manifolds quipped with a (G, M)-structure (given by an atlas of charts in M and transition maps in G). I will introduce the definitions of these two concepts and then sketch the proof of their equivalence as (\infy,1)-categories.

1:00 pm in 345 Altgeld Hall,Tuesday, February 19, 2019

#### Realizations of countable Borel equivalence relations

###### Forte Shinko (Caltech)

Abstract: By a classical result of Feldman and Moore, it is known that every countable Borel equivalence relation can be realized as the orbit equivalence relation of a continuous action of a countable group on a Polish space. However, if we impose further conditions, such as requiring the action to be minimal, then it is no longer clear if such a realization exists. We will detail the progress on characterizing when realizations exist under various conditions, including a complete description in the hyperfinite case. This is joint work with Alexander Kechris.

2:00 pm in 243 Altgeld Hall,Tuesday, February 19, 2019

#### Small Doublings in Abelian Groups of Prime Power Torsion

###### Souktik Roy (Illinois Math)

Abstract: Let $A$ be a subset of $G$, where $G$ is a finite abelian group of torsion $r$. It was conjectured by Ruzsa that if $|A+A|\leq K|A|$, then $A$ is contained in a coset of $G$ of size at most $r^{CK}|A|$ for some constant $C$. The case $r=2$ received considerable attention in a sequence of papers, and was resolved by Green and Tao. Recently, Even-Zohar and Lovett settled the case when $r$ is a prime. In joint work with Yifan Jing (UIUC), we confirm the conjecture when $r$ is a power of prime.

3:00 pm in 243 Altgeld Hall,Tuesday, February 19, 2019

#### Symplectic Springer theory

###### Kevin McGerty (University of Oxford and UIUC)

Abstract: One of the classical results of geometric representation theory is Springer's realization of representations of a Weyl group in the cohomology of the vanishing locus of nilpotent vector fields on the associated flag variety. A rich strain of current research focuses on attempting to extend aspects of Lie theory to the more general context of conical symplectic resolutions''. We will discuss, based on the discovery of Markman and Namikawa that such varieties have a natural analogue of a Weyl group, to what extent one can build an analogue of Springer's theory in this context, recovering for example a construction of Weyl group actions on the cohomology of quiver varieties, first discovered by Nakajima, which unlike previous construction does not require painful explicit verification of the braid relation.

4:00 pm in 243 Altgeld Hall,Tuesday, February 19, 2019

#### Learnability Can Be Undecidable

###### Jacob Trauger (University of Illinois at Urbana–Champaign)

Abstract: This seminar will be on the paper by Shai Ben-David et al, NATURE Mach. Intel. vol 1, Jan 2019, pp 44–48. The author's abstract reads: "The mathematical foundations of machine learning play a key role in the development of the field. They improve our understanding and provide tools for designing new learning paradigms. The advantages of mathematics, however, sometimes come with a cost. Gödel and Cohen showed, in a nutshell, that not everything is provable. Here we show that machine learning shares this fate. We describe simple scenarios where learnability cannot be proved nor refuted using the standard axioms of mathematics. Our proof is based on the fact the continuum hypothesis cannot be proved nor refuted. We show that, in some cases, a solution to the ‘estimating the maximum’ problem is equivalent to the continuum hypothesis. The main idea is to prove an equivalence between learnability and compression."

4:00 pm in 245 Altgeld Hall,Tuesday, February 19, 2019

#### Altgeld-Illini Renovation/Building Project Feedback Session

Abstract: This feedback session will focus on recent renovations done at other math departments. Specific questions that will be discussed at the meeting are:

a. What other math departments have been built or redone in the last 20 years?
b. What is your general impression of each of these spaces?
c. What specific features of particular places are worth copying?

Wednesday, February 20, 2019

3:00 pm in 243 Altgeld Hall,Wednesday, February 20, 2019

#### To Be Announced

###### Elisabeth Moyer (University of Chicago, Geophysical Sciences)

3:00 pm in 2 Illini Hall,Wednesday, February 20, 2019

#### The Geometry of Spectral Curves

###### Matej Penciak (Illinois Math)

Abstract: One way of encoding the data of an integrable system is in terms of the spectral curves. From the curves, it is possible to obtain the constants of motion as integrals over cycles in the curves. In this talk, I will explain some of these classical aspects of integrable systems through some worked out examples. I will also introduce an action-coordinate (AC) duality for integrable systems. I will show how AC duality can be used to relate well-known integrable systems and even construct new integrable systems from old ones. Finally, I hope to describe what the action this AC duality has on spectral curves for some integrable systems of interest.

4:00 pm in 245 Altgeld Hall,Wednesday, February 20, 2019

#### Some necessary uses of logic in mathematics

###### Ilijas Farah (York University)

Abstract: Every now and then, a difficult mathematical problem turns out to be difficult for a particularly objective reason: Provably, it cannot be solved by using 'conventional' means. Some classical examples are proving the Continuum Hypothesis, trisecting an angle, and solving the quintic equation. I’ll discuss more recent examples of such problems, giving some emphasis to the problems arising from the study of operator algebras.

4:00 pm in 343 Altgeld Hall,Wednesday, February 20, 2019

#### Connecting Boolean (un)satisfiability to Graph Theory

###### Vaibhav Karve (Illinois Math)

Abstract: Given a Boolean formula can we find consistent assignments (True or False)for variables such that the formula is satisfied? This is the Boolean Satisfiability problem, a problem of great historic value in computer science. It is the first problem that was proven to be NP-complete. In this talk, I will introduce Satisfiability and explain what the terms P, NP, NP-complete... mean. I will then demonstrate a (surprising)connection between Boolean formulas and graph theory which will help us gain a more visual understanding of when a class of formulas is satisfiable or unsatisfiable. There will be lots of small graphs in this talk.

Thursday, February 21, 2019

11:00 am in 241 Altgeld Hall,Thursday, February 21, 2019

#### Prime number models, large gaps, prime tuples and the square-root sieve

###### Kevin Ford (Illinois Math)

Abstract: We introduce a new probabilistic model for primes, which we believe is a better predictor for large gaps than the models of Cramer and Granville. We also make strong connections between our model, prime k-tuple counts, large gaps and the "square-root sieve". In particular, our model makes a prediction about large prime gaps that may contradict the models of Cramer and Granville, depending on the tightness of a certain sieve estimate. This is joint work with Bill Banks and Terence Tao.

12:00 pm in 243 Altgeld Hall,Thursday, February 21, 2019

#### Taut foliations and left-orderability of 3 manifold groups

###### Ying Hu (University of Nebraska-Omaha)

Abstract: A group G is called left-orderable if there exists a strict total order on G which is invariant under the left-multiplication. Given an irreducible 3-manifold M, it is conjectured that the fundamental group of the 3-manifold is left-orderable if and only if M admits a co-orientable taut foliation. In this talk, we will discuss the left-orderability of the fundamental groups of 3-manifolds that admit co-orientable taut foliations.

2:00 pm in 241 Altgeld Hall,Thursday, February 21, 2019

#### A note on the Liouville function in short intervals

Abstract: We will begin discussing a note of Kaisa Matomaki and Maksym Radziwill on the Liouville function in short intervals. Come prepared to discuss and participate. You can find the note here: https://arxiv.org/abs/1502.02374

2:00 pm in 243 Altgeld Hall,Thursday, February 21, 2019

#### Lipschitz free spaces on finite metric spaces

###### Denka Kutzarova-Ford (UIUC Math)

Abstract: We prove that the Lipschitz free space on any finite metric space contains a large well-complemented subspace which is close to $\ell_1^n$. We show that Lipschitz free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell_1^n$ of the corresponding dimension. These classes contain well-known families of diamond graphs and Laakso graphs. The paper is joint with S. J. Dilworth and M. Ostrovskii.

Friday, February 22, 2019

2:00 pm in 141 Altgeld Hall,Friday, February 22, 2019

#### Monic representations for higher-rank graph C*-algebras

###### Judith Packer (University of Colorado Boulder)

Abstract: We discuss the notion of monic representations for C*-algebras associated to finite higher–rank graphs without sources, generalizing a concept first defined by D. Dutkay and P. Jorgensen for representations of Cuntz algebras. Monic representations are those that, when restricted to the commutative C*-subalgebra of continuous functions on the infinite path space associated to the graph, admit a cyclic vector. We connect these representations to earlier work on dynamical systems with C. Farsi, E. Gillaspy, and S. Kang. The results discussed are based on joint work with C. Farsi, E. Gillaspy, S. Kang, and P. Jorgensen.

3:00 pm in 341 Altgeld Hall,Friday, February 22, 2019

#### Lipschitz Free Spaces

###### Christoper Gartland (Illinois Math)

Abstract: This will be a introduction to Lipschitz free spaces. The Lipschitz free space of a metric space $M$ is a Banach space LF$(M)$ containing $M$ so that for any Banach space $B$ and contractive map $M \to B$, there exists a unique linear contraction LF$(M) \to B$ extending the original map. We'll look at some examples, and discuss current results and open problems.

4:00 pm in 145 Altgeld Hall,Friday, February 22, 2019

#### 27 lines on smooth cubic surfaces

###### Ningchuan Zhang (UIUC)

Abstract: In this talk, I will show that there are $27$ projective lines on a smooth cubic surface in $\mathbb{CP}^3$ by a Chern class computation. This talk is based on a course project I did with Professor Sheldon Katz in Math 524 (now 514) in Spring 2015. No knowledge of algebraic geometry or characteristic classes is assumed.

4:00 pm in 241 Altgeld Hall,Friday, February 22, 2019

#### Functions of Operators

###### Prof. John P D'Angelo   [email] (UIUC Math)

Abstract: We all know what we mean by the derivative operator D = d/dx. What might we mean by the square root of D? In other words, how do we take “half” of a derivative? More generally, how might we take g(D) for a more general function g. Starting from junior high school math (I am not joking!) we figure out the ideas that lead to a nice answer.

4:00 pm in 345 Altgeld Hall ,Friday, February 22, 2019

#### Cancelled

Monday, February 25, 2019

5:00 pm in 241 Altgeld Hall,Monday, February 25, 2019

#### Curvature in Riemannian geometry

Tuesday, February 26, 2019

11:00 am in 345 Altgeld Hall,Tuesday, February 26, 2019

#### What we know so far about "topological Langlands Correspondence"

###### Andrew Salch (Wayne State University)

Abstract: I'll give a survey of some relationships between Galois representations and stable homotopy groups of finite CW-complexes which suggest the possibility of "topological Langlands correspondences." I'll explain what such correspondences ought to be, what their practical consequences are for number theory and for algebraic topology, and I'll explain the cases of such correspondences that are known to exist so far. As an application of one family of known cases, I'll give a topological proof of the Leopoldt conjecture for one particular family of number fields. Some of the results in this talk are joint work with M. Strauch.

12:00 pm in 243 Altgeld Hall,Tuesday, February 26, 2019

#### Congruence subgroups in genus one

###### Autumn Kent (U Wisconsin)

Abstract: I’ll discuss a proof of Asada’s theorem that mapping class groups of punctured tori have the congruence subgroup property.

1:00 pm in Altgeld Hall,Tuesday, February 26, 2019

#### n-dependent groups and fields

Abstract: NIP theories are the first class of the hierarchy of n-dependent structures. The random n-hypergraph is the canonical object which is n-dependent but not (n-1)-dependent. Thus the hierarchy is strict. But one might ask if there are any algebraic objects (groups, rings, fields) which are strictly n-dependent for every n? We will start by introducing the n-dependent hierarchy and present all known results on n-dependent groups and fields. These were (more or less) inspired by the above question.

2:00 pm in 243 Altgeld Hall,Tuesday, February 26, 2019

#### 2-connected hypergraphs with no long cycles

###### Ruth Luo (Illinois Math)

Abstract: The Erdős–Gallai theorem gives an upper bound for the maximum number of edges in an $n$-vertex graph with no cycle of length $k$ or longer. Recently, many analogous results for $r$-uniform hypergraphs with no Berge cycle of length $k$ or longer have appeared. In this talk, we present a result for $2$-connected hypergraphs without long Berge cycles. For $n$ large with respect to $r$ and $k$, our bound is sharp and is significantly stronger than the bound without restrictions on connectivity. This is joint work with Zoltán Füredi and Alexandr Kostochka.

3:00 pm in 243 Altgeld Hall,Tuesday, February 26, 2019

#### Pure cohomology of multiplicative quiver varieties

###### Thomas Nevins (UIUC)

Abstract: Multiplicative quiver varieties are certain quasiprojective algebraic varieties, defined by Crawley-Boevey and Shaw, associated to quivers. Examples include many moduli spaces of surface group representations (with punctures), a.k.a. moduli spaces of connections on punctured surfaces. I will introduce the basics of these varieties and explain joint work with McGerty that describes generators of the Hodge-theoretically "pure" part of their cohomology rings.

4:00 pm in 245 Altgeld Hall,Tuesday, February 26, 2019

#### Altgeld-Illini Renovation/Building Project Feedback Session

Abstract: This feedback session will focus on office space. Everyone is encouraged attend and the committee would like to hear from graduate students, non-tenure track instructors and lecturers and postdocs, in addition to faculty.

Wednesday, February 27, 2019

3:00 pm in 2 Illini Hall,Wednesday, February 27, 2019

#### Dieudonné crystals associated to formal groups

###### Ningchuan Zhang (Illinois Math)

Abstract: In this talk, I will introduce Dieudonné crystals associated to commutative formal group schemes. The focus of this talk will be on the construction of the contravariant Dieudonné crystal functor and explicit computation of some examples. I'll also mention its relation with extensions and deformations of formal groups if time allows.

3:00 pm in 341 Altgeld Hall,Wednesday, February 27, 2019

#### "Generic representations of abelian groups and extreme amenability" by J. Melleray and T. Tsankov (Part 2)

###### Dakota Ihli (UIUC)

Abstract: In this series of talks, we discuss the paper in the title [arXiv link].

3:00 pm in 243 Altgeld Hall,Wednesday, February 27, 2019

#### To Be Announced

###### Sooin Yun (University of Illinois, Statistics)

4:00 pm in 245 Altgeld Hall,Wednesday, February 27, 2019

###### Marissa Loving (Illinois Math)

Abstract: In this talk, I will tell the story of my journey through grad school. I will attempt to be as blunt as possible about the ups and downs I have experienced and touch on some of the barriers I have encountered (both internally and externally). If you have ever felt like you don’t belong or worried that you have made others feel that way, this talk is for you.

Thursday, February 28, 2019

11:00 am in 241 Altgeld Hall,Thursday, February 28, 2019

#### Using q-analogues to transform singularities

###### Kenneth Stolarsky (Illinois Math)

Abstract: This is a mostly elementary talk about polynomials and their q-analogues, filled with conjectures based on numerical evidence. For example, if ( x - 1 ) ^ 4 is replaced by a q-analogue, what happens to the root at x = 1 ? These investigations accidentally answer a question posed by J. Browkin about products of roots that was also answered by Schinzel some decades ago. We also look at how certain q-analogues are related to each other.

4:00 pm in 245 Altgeld Hall,Thursday, February 28, 2019

#### Quivers, representation theory and geometry

###### Kevin McGerty (University of Oxford and Visiting Fisher Professor, University of Illinois)

Abstract: A quiver is an oriented graph. It has a natural algebra associated to it called the path algebra, which as the name suggests has a basis given by paths in the quiver with multiplication given by concatenation. The representation theory of these algebras encompasses a number of classical problems in linear algebra, for example subspace arrangements and Jordan canonical form. A remarkable discovery of Gabriel however in the 1970s revealed a deep connection between these algebras and Lie theory, which has subsequently lead to a rich interaction between quivers, Lie theory and algebraic geometry. This talk will begin by outlining the elementary theory of representations of path algebras, explain Gabriel's result and survey some of the wonderful results which it has led to in Lie theory: the discovery of the canonical bases of quantum groups, the geometric realization of representations of affine quantum groups by Nakajima, and most recently deep connections between representations of symplectic reflection algebras and affine Lie algebras.

Friday, March 1, 2019

2:00 pm in 141 Altgeld Hall,Friday, March 1, 2019

#### Poisson equation, its approximation, and error analysis

###### Amir Taghvaei (Illinois MechSE)

Abstract: In this talk, I discuss the computational problem of approximating the solution of a probability weighted Poisson equation, in terms of finite number of particles sampled from the probability distribution. The poisson equation arises in the theory of nonlinear filtering and optimal transportation. I present an approximation procedure based on the stochastic viewpoint of the problem. Then, I present the error analysis of the approximation using the Lyapunov stability theory in stochastic analysis.

4:00 pm in 345 Altgeld Hall ,Friday, March 1, 2019

#### "The complexity of topological group isomorphism" by A. Kechris, A. Nies, and K. Tent (Part 3)

###### Mary Angelica Gramcko-Tursi (UIUC)

Abstract: This will be the third talk of the series on the paper in the title [arXiv link], which deals with the classification of some natural classes of non-Archimedean groups (= closed subgroups of S) up to topological group isomorphism. It gives a general criterion for a class of non-Archimedean groups to show that the topological group isomorphism on it is Borel-classifiable by countable structures. This criterion is satisfied by the classes of profinite groups, locally compact non-Archimedean groups, and oligomorphic groups. In this talk, we will show that one or two of the aforementioned classes satisfy this criterion.

4:00 pm in 145 Altgeld Hall,Friday, March 1, 2019

#### Exposition on motives

###### Tsutomu Okano (UIUC)

Abstract: The proof of Weil conjectures led Grothendieck to think about categories of motives. This is supposed to be an abelian category that contains all the arithmetic-geometric information of varieties. Such a category has not yet been proved to exist. However, there are convincing partial answers which I hope to communicate in this talk. I will describe Grothendieck's construction of pure Chow motives, then Voevodsky's construction of the conjectured derived category of motives. Towards the end, I will describe the connection with motivic homotopy theory.

Saturday, March 2, 2019

4:00 pm in Altgeld Hall,Saturday, March 2, 2019

#### To Be Announced

Monday, March 4, 2019

5:00 pm in 241 Altgeld Hall,Monday, March 4, 2019

#### Curvatures of Riemannian Lie groups

Tuesday, March 5, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, March 5, 2019

#### Some recent progress on the Falconer distance conjecture and applications

###### Alex Iosevich (U. Rochester)

Abstract: We are going to discuss some recent results related to the Falconer distance conjecture and applications of some of these methods to the theory of exponential bases and frames.

1:00 pm in 345 Altgeld Hall,Tuesday, March 5, 2019

#### Descriptive graph combinatorics with applications to geometry

###### Spencer Unger (Tel Aviv University)

Abstract: The Banach–Tarski paradox states that (assuming the axiom of choice) a unit ball in $\mathbb{R}^3$ can be partitioned into $5$ sets which can be rearranged by isometries to partition two unit balls. This famous result is part of a larger line of early 20th century research which sought to understand the relation between foundations of measure theory and generalizations of classical ideas such as decomposing polygons into congruent sets.
In the last few years, there has been a resurgence of interest in these geometrical paradoxes. These results have the unifying theme that the "paradoxical" sets in many classical geometrical paradoxes can surprisingly be much "nicer" than one would naively expect. In this talk, we give a survey of these results and explain a few of the ideas that go in to a constructive solution to Tarski's circle squaring problem. This is joint work with Andrew Marks.

2:00 pm in 243 Altgeld Hall,Tuesday, March 5, 2019

#### Polynomial to exponential transition in hypergraph Ramsey theory

###### Lina Li (Illinois Math)

Abstract: Let $r_k(s, t; n)$ be the minimum $N$ such that every red/blue colorings of the edges of $K^k_N$ contains a blue $K^k_n$ or has $s$ vertices which induce at least $t$ red edges. The study of $r_k(s, t; n)$ is related to many other classical problems, such as classical Ramsey theory and Erdős–Szekeres problem.

The main problem of Erdős and Hajnal asks for the growth rate of $r_k(s, t; n)$ when $t$ changes from $1$ to $s \choose k$. In particular, they conjectured that for given $s$ and $k$, the threshold of $t$ which separates the polynomial growth rate and super polynomial growth rate can be calculated precisely by a recursive formula.

In this talk, I will present the history of this problem, and discuss the most recent progress made by Mubayi and Razborov, who resolve the above conjecture.

3:00 pm in 243 Altgeld Hall,Tuesday, March 5, 2019

#### Modeling Learning and Strategy Formation in Phase Transitions in Cortical Networks

###### Kesav Krishnan (University of Illinois at Urbana–Champaign)

Abstract: In the first of 2 seminars on this paper by Kozma++ we review the experimental data and their graph-theoretic methods. In the second, we review the mathematical details and offer a critique of their results. Here is a paraphrase of the authors abstract: Learning in mammalian brains is commonly modeled in terms of synaptic connections in a cortical network and the formation of limit cycle oscillators of a dynamical system. Learning is inferred by the re-emergence of the oscillatory regimes by repeating the stimulus. Here the authors use random graphs and boostrap percolation with excitatory and ihibitory vertices. The phase transition from fixed point attractors to limit cycles (Hopf bifurcations) represent changes in cortical networks during category learning. A correspondence with analogous event in the gerbil cortex is based on experiments with electro-cortiographs (ECoG) arrays. They discuss how learning leads to categorization and strategy formation, and how the theoretical modeling results can be used for designing learning and adaptation in computationally aware intelligent machines.

4:00 pm in 245 Altgeld Hall,Tuesday, March 5, 2019

#### Altgeld-Illini Renovation/Building Project Feedback Session

Abstract: CANCELED

6:30 pm in 1320 Digital Computer Lab,Tuesday, March 5, 2019

#### Introduction to CAS, Ratemaking, and Emerging Insure Tech Trends

###### Steve Armstrong (Allstate; President-elect, the Casualty Actuarial Society)

Abstract: Steve Armstrong is a personal insurance expert with over 25 years of extensive experience in pricing, product design, underwriting, and regulatory work. Throughout this career, Steve has led teams of actuaries, predictive modelers, and product analysts; he has also served as an expert witness in several states. Steve began his career at Allstate Insurance Company, where he worked as an actuary overseeing actuarial and analytical talent for auto and homeowners insurance focusing on both the state-specific rate reviews and filings and the predictive modeling and countrywide rating plan development. During his tenure, Steve was responsible for introducing the product and rating algorithm for Drivewise, Allstate’s usage-based insurance endeavor before he left in 2012. Between 2012 and 2017, Steve worked for two large global insurance companies to help bring actuarial best practices to these companies and to other countries ar ound the world. Steve recently returned to Allstate to oversee actuarial work and lead actuarial pricing strategy for private passenger automobile. Steve received his Bachelor of Science degree in Actuarial Science from the University of Illinois – Urbana-Champaign in 1992 and his MBA from the University of Illinois – Chicago in 2003. Steve received his Casualty Actuarial Society (CAS) Fellowship in 1996; he served on the CAS Board of Directors from 2011- 2014 and as the CAS Vice President of Admissions from 2014-2017.

Wednesday, March 6, 2019

3:00 pm in 2 Illini Hall,Wednesday, March 6, 2019

#### Abelian Varieties in Positive Characteristic

###### Ravi Donepudi (Illinois Math)

Abstract: This talk will be an introduction to the theory of abelian varieties over fields of positive characteristic. The presence of the non-separable Frobenius automorphism in this context gives the theory a flavor entirely different from over the complex numbers. An important question in this area is to characterize which abelian varieties (with extra data) arise as Jacobians of smooth curves. Much of the progress on this problem has been through studying some stratifications of moduli spaces of abelian varieties. We will introduce these moduli spaces and stratifications, and survey interesting results in this area.

3:00 pm in 243 Altgeld Hall,Wednesday, March 6, 2019

#### To Be Announced

###### Ben Vega Westhoff (University of Illinois, Atmospheric Sciences)

3:00 pm in 341 Altgeld Hall,Wednesday, March 6, 2019

#### Abstract systems of congruences

###### Spencer Unger (Tel Aviv University)

Abstract: Abstract systems of congruences provide a different perspective for viewing geometrical paradoxes from the early 20th century. Consider partitioning a space into $n$ pieces $A_1, A_2, \dots, A_n$. An abstract system of congruences is a collection of statements (called congruences), like $A_2 \cup A_6$ is isometric to $A_{17}$. Such a system is satisfied by a particular partition if each congruence is satisfied. We survey some recent results and some open problems.

Thursday, March 7, 2019

11:00 am in 241 Altgeld Hall,Thursday, March 7, 2019

#### Diophantine problems and a p-adic period map

###### Brian Lawrence (University of Chicago)

Abstract: I will outline a proof of Mordell's conjecture / Faltings's theorem using p-adic Hodge theory. I'll start with a discussion of cohomology theories in algebraic geometry, and build from there. The paper is joint with Akshay Venkatesh.

12:00 pm in 243 Altgeld Hall,Thursday, March 7, 2019

#### Atoroidal dynamics of subgroups of $$Out(F_N)$$

###### Caglar Uyanik (Yale)

Abstract: I will discuss several examples to illustrate how the dynamics of the $$Out(F_N)$$ action on various spaces reflects on the algebraic structure of the $$Out(F_N)$$ itself. In particular, I will talk about a new subgroup classification theorem for $$Out(F_N)$$ which is joint work with Matt Clay.

4:00 pm in 245 Altgeld Hall,Thursday, March 7, 2019

#### New examples of Calabi-Yau metrics on a complex vector space

###### Frederic Rochon (University of Quebec in Montreal)

Abstract: After reviewing how the Riemann curvature tensor describes the local geometry of a space and how it may reflect some global aspects of its topology, we will focus on a special type of geometry: Calabi-Yau manifolds. By smoothing singular Calabi-Yau cones and using suitable compactifications by manifolds with corners, we will explain how to construct new examples of complete Calabi-Yau metrics on a complex vector space. Our examples are of Euclidean volume growth, but with tangent cone at infinity having a singular cross-section. This is a joint work with Ronan J. Conlon.

Friday, March 8, 2019

2:00 pm in 141 Altgeld Hall,Friday, March 8, 2019

#### On generic monothetic subgroups of Polish groups

###### Dakota Thor Ihli (Illinois Math)

Abstract: Given a topological group $G$, we ask whether the group $\overline{\left\langle g \right\rangle}$ has the same isomorphism type for "most" $g \in G$. More precisely, is there a group $H$ such that the set $\left\{ g \in G : \overline{\left\langle g \right\rangle} \cong H \right\}$ is dense? Comeagre? If so, can we identify this $H$? In this expository talk I will discuss known results and conjectures for certain Polish groups. Emphasis will be given to the case when $G$ is the group of Lebesgue-measure preserving automorphisms of the unit interval.

3:00 pm in 341 Altgeld Hall,Friday, March 8, 2019

#### Completely bounded analogues of the Choquet and Shilov boundaries for operator spaces

###### Raphael Clouatre (University of Manitoba)

Abstract: Given a unital operator algebra, it is natural to seek the smallest $C^*$-algebra generated by a completely isometric image of it, by analogy with the classical Shilov boundary of a uniform algebra. In keeping with this analogy, one method for constructing the so-called $C^*$-envelope is through a non-commutative version of the Choquet boundary. It is known that such a procedure can be also be applied to operator spaces, although in this case the envelope has less structure. In this talk, I will present a certain completely bounded version of the non-commutative Choquet boundary of an operator space that yields the structure of a $C^*$-algebra for the associated Shilov boundary. I will explain how the resulting $C^*$-algebras enjoy some of the properties expected of an envelope, but I will also highlight their shortcomings along with some outstanding questions about them. This is joint work with Christopher Ramsey.

4:00 pm in 145 Altgeld Hall,Friday, March 8, 2019

#### Basics of Chern Simons Theory

###### Yidong Chen (UIUC)

Abstract: In this talk I'll explain Atiyah's "axioms" for topological field theory and construct two examples: Chern Simons theory with finite group over any compact oriented manifold, and Chern Simons theory with compact simply connected Lie group over a compact connected 3-manifold. The latter (with SU(2)) is the quintessential example for Chern Simons theory in the physics literature.

4:00 pm in 347 Altgeld Hall,Friday, March 8, 2019

#### When a Prime Number Ceases to be Prime

###### Ravi Donepudi   [email] (UIUC Math)

Abstract: Primes are commonly defined as those numbers whose only factors are 1 and themselves. This assumes that we only allow integers in their factorization. What happens if we allow fractions as factors or even irrational numbers? Will certain primes lose their status as "primes"? Will new "primes" be born to take their place? What does being prime even mean anymore? We will answer these and other questions which lead us to the exciting field of algebraic number theory.

4:00 pm in 345 Altgeld Hall ,Friday, March 8, 2019

#### Organizational meeting

Monday, March 11, 2019

9:00 amMonday, March 11, 2019

Abstract: Visiting day for admitted PhD students.

2:00 pm in 245 Altgeld Hall,Monday, March 11, 2019

#### A brief survey of extremal combinatorics and some new results for (hyper)graphs

###### Ruth Luo (Illinois Math)

Abstract: Extremal combinatorics is a branch of discrete mathematics which studies how big or how small a structure (e.g., a graph, a set of integers, a family of sets) can be given that it satisfies some set of constraints. Extremal combinatorics has many applications in fields such as number theory, discrete geometry, and computer science. Furthermore, methods in extremal combinatorics often borrow tools from other fields such as algebra, probability theory, and analysis. In this talk, we will discuss some benchmark results in the field as well as some recent results for extremal problems in graphs and hypergraphs.

5:00 pm in 241 Altgeld Hall,Monday, March 11, 2019

#### Riemannian geometry and Clifford algebras

###### Nijholt, Eeltje Cornelis (UIUC)

Tuesday, March 12, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, March 12, 2019

#### Hyperfiniteness and descriptive combinatorics

###### Clinton Conley (Carnegie Mellon)

Abstract: We survey some recent results on connections between descriptive set-theoretic properties of Borel graphs and hyperfiniteness of their connectedness equivalence relation. For convenience, we will focus on chromatic numbers with various measurability constraints. This talk will include joint work with Jackson, Marks, Miller, Seward, Tucker-Drob.

1:00 pm in 347 Altgeld Hall,Tuesday, March 12, 2019

#### The lattice bump multiplier problem

###### Loukas Grafakos (University of Missouri-Columbia)

Abstract: Given a smooth bump supported in a ball centered at the origin in $R^n$, we consider the multiplier formed by adding the translations of this bump by $N$ distinct lattice points. We investigate the behavior as $N$ tends to infinity of the $L^p$ norm of the multiplier operators associated with this finite sum of $N$ bumps.

2:00 pm in 243 Altgeld Hall,Tuesday, March 12, 2019

#### Learning on hypergraphs: spectral theory and clustering with applications

###### Pan Li (Illinois ECE)

Abstract: Learning on graphs is an important problem in machine learning, computer vision, and data mining. Traditional algorithms for learning on graphs primarily take into account only low-order connectivity patterns described at the level of individual vertices and edges. However, in many applications, high-order relations among vertices are necessary to properly model a real-life problem. In contrast to the low-order cases, in-depth algorithmic and analytic studies supporting high-order relations among vertices are still lacking. To address this problem, we introduce a new mathematical model family, termed inhomogeneous hypergraphs, which captures the high-order relations among vertices in a very extensive and flexible way. Specifically, as opposed to classic hypergraphs that treats vertices within a high-order structure in a uniform manner, inhomogeneous hypergraphs allow one to model the fact that different subsets of vertices within a high-order relation may have different structural importance. We propose a series of algorithmic and analytic results for this new model, including inhomogeneous hypergraph clustering, spectral hypergraph theory, and novel applications ranging from food-web and ranking analysis to subspace segmentation. All proposed algorithms come with provable performance guarantees and are evaluated on real datasets; the results demonstrate significant performance improvements compared to classical learning algorithms.

3:00 pm in 243 Altgeld Hall,Tuesday, March 12, 2019

#### To Be Announced

###### TBA

4:00 pm in 245 Altgeld Hall,Tuesday, March 12, 2019

#### Altgeld-Illini Renovation/Building Project Feedback Session

Abstract: This feedback session will focus on the Math Library.

Wednesday, March 13, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, March 13, 2019

#### Hyperfiniteness and descriptive combinatorics: ideas and proofs

###### Clinton Conley (Carnegie Mellon)

Abstract: We discuss some ideas and proofs behind the results surveyed in the first part of this talk on Tuesday.

3:00 pm in 243 Altgeld Hall,Wednesday, March 13, 2019

#### To Be Announced

###### Trevor Harris (University of Illinois, Statistics)

3:00 pm in 2 Illini Hall,Wednesday, March 13, 2019

#### What are matrix factorizations?

###### Jesse Huang (Illinois Math)

Abstract: A matrix factorization is, roughly speaking, what looks like AB=fId where f is a polynomial and every square matrix in the equation takes value in the polynomial ring. This notion was originally introduced in the study of homological algebra on (singular) complete intersections and then generalized and made into a younger sibling of the derived category of coherent sheaves. The state-of-the-art consolidates the study of things like hypersurface singularities and (A to B) mirror symmetry for non-CYs. I will try to showcase some basics and survey through a handful of well-known results in this talk.

5:00 pm in 314, 245, 243 Altgeld Hall,Wednesday, March 13, 2019

#### Mid Semester Meeting

Abstract: The IGL Mid-Semester meeting will be taking place next Wednesday evening (03/13) from 5-6:30 pm in Altgeld Hall. Due to the large number of IGL projects, this mid-semester meeting will involve three parallel sessions, with each group presenting in one of the following three rooms as indicated below. Furthermore, the presentations will allowed to be a bit longer: this semester each groups presentation will last up to 7 minutes, plus a minute for questions to occur. We expect each group to last roughly 10 minutes.

AH 314:
Natural Selection and the Bystander Effect
Modeling Prevalence of Juul and other E-Cigarette
Use Evaluating models of social group competition
Do Blue Skies drive away Pollution?
The Smart Foodie
San Francisco Parking
Movement Disorder Gait Data
Uber Air Taxis
Virtual Reality and Movement Disorders

AH 245:
Pairs of Disjoint Matchings
Homological Algebra of Quiver Representations
Automata and Numeration Systems
Developing Exciting Outreach Material Decomposition
Theorems for Spectra
Finite Reflection Groups and Related Topics
Continuous Factorization of the Identity Matrix

AH 243:
Simulating Multi-Soliton Solutions to NLS and KdV
Talbot Effect for Dispersive Partial Differential Equations
Interactive Tools for Integrable Dynamical Systems
Interactive Visualizations in Calculus and Probability
Problems on Markov Chains arising from Operator Algebras
Natural Extension Domains of alpha-odd continued fractions
Bounds and Optimizations for Distributed Storage
Search for New Tensegrity Configurations

Thursday, March 14, 2019

11:00 am in 241 Altgeld Hall,Thursday, March 14, 2019

#### Extremal primes for elliptic curves without complex multiplication

###### Ayla Gafni (Rochester Math)

Abstract: Fix an elliptic curve $E$ over $\mathbb{Q}$. An ''extremal prime'' for $E$ is a prime $p$ of good reduction such that the number of rational points on $E$ modulo $p$ is maximal or minimal in relation to the Hasse bound. In this talk, I will discuss what is known and conjectured about the number of extremal primes $p\le X$, and give the first non-trivial upper bound for the number of such primes when $E$ is a curve without complex multiplication. The result is conditional on the hypothesis that all the symmetric power $L$-functions associated to $E$ are automorphic and satisfy the Generalized Riemann Hypothesis. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in recent work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.

12:00 pm in 243 Altgeld Hall,Thursday, March 14, 2019

###### Pierre Will (Institut Fourier)

Abstract: In this talk, I will explain how it is possible to construct interesting geometric structures modelled on the boundary at infinity of the complex hyperbolic 2-space. In particular, I will describe examples of hyperbolic 3-manifolds that appear this way. This talk is based on joint works with Antonin Guilloux, and John Parker.

12:30 pm in 464 Loomis,Thursday, March 14, 2019

#### A proposal for nonabelian mirrors in two-dimensional theories

###### Eric Sharpe (Virginia Tech)

Abstract: In this talk we will describe a proposal for nonabelian mirrors to two-dimensional (2,2) supersymmetric gauge theories, generalizing the Hori-Vafa construction for abelian gauge theories. By applying this to spaces realized as symplectic quotients, one can derive B-twisted Landau-Ginzburg orbifolds whose classical physics encodes quantum cohomology rings of those spaces. The proposal has been checked in a variety of cases, but for sake of time the talk will focus on exploring the proposal in the special case of Grassmannians.

2:00 pm in 243 Altgeld Hall,Thursday, March 14, 2019

#### Generalized Derivatives

###### Alastair Fletcher (Northern Illinois University)

Abstract: Quasiregular mappings are only differentiable almost everywhere. There is, however, a satisfactory replacement for the derivative at points of non-diffferentiability. These are generalized derivatives and were introduced by Gutlyanskii et al in 2000. In this talk, we discuss some recent results on generalized derivatives, in particular the question of how many generalized derivatives there can be at a particular point, and explaining how versions of the Chain Rule and Inverse Function Formula hold in this setting. We also give some applications to Schroeder functional equations.

3:00 pm in 243 Altgeld Hall,Thursday, March 14, 2019

#### To Be Announced

###### Satya Mandal (University of Kansas)

Abstract: Title: Splitting property of projective modules, by Homotopy obstructions Speaker: Satya Mandal, U. of Kansas \noindent{\bf Abstract:} Follow the link: http://mandal.faculty.ku.edu/talks/abstractIllinoisMarch19.pdf Alternate version: The theory of vector bundles on compact hausdorff spaces $X$, guided the research on projective modules over noetherian commutative rings $A$. There has been a steady stream of results on projective modules over $A$, that were formulated by imitating existing results on vector bundles on $X$. The first part of this talk would be a review of this aspects of results on projective modules, leading up to some results on splitting projective $A$-modules $P$, as direct sum $P\cong Q\oplus A$. % Our main interest in this talk is to define an obstruction class $\varepsilon(P)$ in a suitable obstruction set (preferably a group), to be denoted by $\pi_0\left({\mathcal LO}(P) \right)$. Under suitable smoothness and other conditions, we prove that $$\varepsilon(P)\quad {\rm is~trivial~if~and~only~if}~ P\cong Q\oplus A$$ Under similar conditions, we prove $\pi_0\left({\mathcal LO}(P) \right)$ has an additive structure, which is associative, commutative and has n unit (a "monoid").

Friday, March 15, 2019

4:00 pm in 345 Altgeld Hall ,Friday, March 15, 2019

#### The theory of addition with predicates for the powers of 2 and 3

###### Christian Schulz (UIUC Math)

Abstract: This talk concerns the intricate boundary between decidable and undecidable of expansions of Presburger artithmetic, i.e., the structure $(\mathbb{N}, +)$. For a natural number $p \ge 2$, let $p^{\mathbb{N}}$ denote the set of powers of $p$, and let $V_p$ be a predicate that allows us to access the full base-$p$ expansion of a natural number. It is known that the expansion $(\mathbb{N}, +, V_p)$ of Presburger arithmetic retains decidability, but $(\mathbb{N}, +, V_p, q^{\mathbb{N}})$, for $q$ multiplicatively independent from $p$, has an undecidable theory. In this talk, I present a proof that the reduct $(\mathbb{N}, +, p^{\mathbb{N}}, q^{\mathbb{N}})$ also has an undecidable theory, specifically in the case $p = 2$, $q = 3$. I conclude with a note on how the proof extends to other structures, as well as some discussion of directions for further research.

4:00 pm in 145 Altgeld Hall,Friday, March 15, 2019

#### Some aspects of Foliations of 3-manifolds

###### Gayana Jayasinghe (UIUC)

Abstract: While foliations have proven to be a useful tool for studying the topology and geometry of manifolds, in lower dimensions, they allow one to create and admire extremely beautiful pictures. Renowned masters of this art such as William Thurston and David Gabai have developed a many-layered machinery to manipulate and construct "nice" foliations. I will assume very little knowledge and will introduce the basics, then talk about some things I found interesting. My props will be edible versions of these you can study at your leisure.

Tuesday, March 19, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, March 19, 2019

#### The Steklov and Laplacian spectra of Riemannian manifolds with boundary

###### Alexandre Girouard   [email] (Université Laval)

Abstract: The Dirichlet-to-Neumann map is a first order pseudodifferential operator acting on the smooth functions of the boundary of a compact Riemannian manifold M. Its spectrum is known as the Steklov spectrum of M. The asymptotic behaviour (as j tends to infinity) of the Steklov eigenvalues s_j is determined by the geometry of the boundary of M. Neverthless, each individual eigenvalue can become arbitrarily big if the Riemannian metric is perturbed adequately. This can be achieved while keeping the geometry of the boundary unchanged, but it requires wild perturbations in arbitrarily small neighborhoods of the boundary. In recent work with Bruno Colbois and Asma Hassannezhad, we impose constraints on the geometry of M on and near its boundary. This allows the comparison of each Steklov eigenvalue s_j with the corresponding eigenvalues l_j of the Laplace operator acting on the boundary. This control is uniform in the index j. The proof is based on a generalized Pohozaev identity and on comparison results for the principal curvatures of hypersurfaces that are parallel to the boundary.

Monday, March 25, 2019

5:00 pm in 241 Altgeld Hall,Monday, March 25, 2019

#### Riemannian geometry and Clifford algebras II

###### Nijholt, Eeltje Cornelis (UIUC)

Tuesday, March 26, 2019

1:00 pm in Altgeld Hall,Tuesday, March 26, 2019

#### To Be Announced

1:00 pm in 345 Altgeld Hall,Tuesday, March 26, 2019

#### Cancelled

2:00 pm in 243 Altgeld Hall,Tuesday, March 26, 2019

#### Linearity of Saturation for Berge Hypergraphs

###### Sean English (Ryerson University)

Abstract: For a graph $F$, we say a hypergraph $H$ is Berge-$F$ if it can be obtained from $F$ be replacing each edge of $F$ with a hyperedge containing it. We say a hypergraph is Berge-$F$-saturated if it does not contain a Berge-$F$, but adding any hyperedge creates a copy of Berge-$F$. The $k$-uniform saturation number of Berge-$F$, $\mathrm{sat}_k(n,\text{Berge-}F)$ is the fewest number of edges possible over all Berge-$F$-saturated $k$-uniform hypergraphs on $n$ vertices.

In this talk we will explore some specific saturation numbers for Berge hypergraphs. We will also see that at least for small uniformities, these numbers grow linearly with $n$, extending a classical result of Kászonyi and Tuza. Finally, we will mention many interesting open problems in this area of research.

3:00 pm in 243 Altgeld Hall,Tuesday, March 26, 2019

#### No Seminar This Week

###### TBA

4:00 pm in 245 Altgeld Hall,Tuesday, March 26, 2019

#### Altgeld-Illini Renovation/Building Project Feedback Session

Abstract: This feedback session will focus on classrooms and technology

Wednesday, March 27, 2019

3:00 pm in 2 Illini Hall,Wednesday, March 27, 2019

#### Intersection Theory I - Rational Equivalence

###### Martino Fassina (Illinois Math)

Abstract: This is the first talk for our reading group on Intersection Theory. The material presented roughly corresponds to Chapter 1 of Fulton's book. I will introduce concepts such as cycles, rational equivalence, proper pushforwards and flat pullbacks. The focus will be on intuition and explicit examples.

3:00 pm in 243 Altgeld Hall,Wednesday, March 27, 2019

#### To Be Announced

###### Matthew Huber (Purdue University, Earth, Atmospheric and Planetary Sciences)

3:00 pm in 341 Altgeld Hall,Wednesday, March 27, 2019

#### "Generic representations of abelian groups and extreme amenability" by J. Melleray and T. Tsankov (Part 3)

###### Dakota Ihli (UIUC)

Abstract: In this series of talks, we discuss the paper in the title [arXiv link]. In this final talk, we prove that the set of probability measure preserving automorphisms that topologically generate a copy of the group $L_0(\mathbb{T})$ is dense in $\mathrm{Aut}(\mu)$.

Thursday, March 28, 2019

11:00 am in 241 Altgeld Hall,Thursday, March 28, 2019

#### Core partitions, Numerical semigroups, and Polytopes

###### Hayan Nam (University of California at Irvine)

Abstract: A partition is an $a$-core partition if none of its hook lengths are divisible by $a$. It is well known that the number of $a$-core partitions is infinite and the number of simultaneous $(a, b)$-core partitions is a generalized Catalan number if $a$ and $b$ are relatively prime. Numerical semigroups are additive monoids that have finite complements, and they are closely related to core partitions. The first half of the talk, we will talk about an expression for the number of simultaneous $(a_1,a_2,\dots, a_k)$-core partitions. In the second half, we discuss the relationship between numerical semigroups and core partitions, along with how to count numerical semigroups with certain restrictions.

12:00 pm in 243 Altgeld Hall,Thursday, March 28, 2019

#### Classifying incompressible surfaces in hyperbolic mapping tori

###### Sunny Xiao (Brown U)

Abstract: One often gains insight into the topology of a manifold by studying its sub-manifolds. Some of the most interesting sub-manifolds of a 3-manifold are the "incompressible surfaces", which, intuitively, are the properly embedded surfaces that can not be further simplified while remaining non-trivial. In this talk, I will present some results on classifying orientable incompressible surfaces in a hyperbolic mapping torus whose fibers are 4-punctured spheres. I will explain how such a surface gives rise to a path which satisfies certain combinatorial properties in the arc complex of the 4-punctured sphere. This extends and generalizes results of Floyd, Hatcher, and Thurston.

2:00 pm in 241 Altgeld Hall,Thursday, March 28, 2019

#### Joint Shapes of Quartic Fields and Their Cubic Resolvents

###### Piper Harron (University of Hawaii)

Abstract: In studying the (equi)distribution of shapes of quartic number fields, one relies heavily on Bhargava's parametrizations which brings with it a notion of resolvent ring. Maximal rings have unique resolvent rings so it is possible to live a long and healthy life without understanding what they are. The authors have decided, however, to forsake such bliss and look into what ever are these rings and what happens if we consider their shapes along with our initial number fields. What indeed! Please stay tuned. (Joint with Christelle Vincent)

3:00 pm in 347 Altgeld Hall,Thursday, March 28, 2019

#### Complexity, Combinatorial Positivity, and Newton Polytopes

###### Colleen Robichaux   [email] (UIUC)

Abstract: The nonvanishing problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby, in amenable cases, nonvanishing is in the complexity class ${\sf NP}\cap {\sf coNP}$ of problems with "good characterizations''. This suggests a new algebraic combinatorics viewpoint on complexity theory. This paper focuses on the case of Schubert polynomials. These form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. We give a tableau criterion for nonvanishing, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the $n\times n$ grid, together with a theorem of A. Fink, K. Meszaros, and A. St. Dizier (2018), which proved a conjecture of C. Monical, N. Tokcan, and A. Yong (2017). This is joint work with Anshul Adve and Alexander Yong.

4:00 pm in 245 Altgeld Hall,Thursday, March 28, 2019

#### Spherical conical metrics

###### Xuwen Zhu (University of California Berkeley)

Abstract: The problem of finding and classifying constant curvature metrics with conical singularities has a long history bringing together several different areas of mathematics. This talk will focus on the particularly difficult spherical case where many new phenomena appear. When some of the cone angles are bigger than $2\pi$, uniqueness fails and existence is not guaranteed; smooth deformation is not always possible and the moduli space is expected to have singular strata. I will give a survey of several recent results regarding this singular uniformization problem, connecting PDE techniques with complex analysis and synthetic geometry. Based on joint works with Rafe Mazzeo and Bin Xu.

Friday, March 29, 2019

3:00 pm in 245 Altgeld Hall,Friday, March 29, 2019

#### How to Become a Liberated Mathematician in 13+3 Painful Years

###### Piper Harron (University of Hawai'i at Mānoa )

Abstract: Piper H never wanted to be liberated. She would have much preferred to be conventionally successful, living by other people's standards. Though she tried, she couldn't make herself fit. You can say she has some complaints. Some people want to spread her message, other people think she needs to go away forever. In this talk Piper lets you in on the secret that actually she's just a very tired person trying to find more time for naps.

4:00 pm in 145 Altgeld Hall,Friday, March 29, 2019

#### Geometric ideas in number theory

###### Robert Dicks (UIUC)

Abstract: Jurgen Neukirch in 1992 wrote that Number Theory is Geometry. At first glance, it seems nothing could be further from the truth, but it turns out that tools such as vector bundles, cohomology, sheaves, and schemes have become indispensable for understanding certain chapters of number theory in recent times. The speaker aims to discuss an analogue in the context of number fields of the classical Riemann-Roch theorem, which computes dimensions of spaces of meromorphic functions on a Riemann surface in terms of its genus. The aim is for the talk to be accessible for any graduate student; we'll find out what happens.

4:00 pm in 345 Altgeld Hall ,Friday, March 29, 2019

#### Generalized sum-product phenomenon for polynomials

###### Souktik Roy (UIUC Math)

Abstract: Suppose $P(x,y)$ and $Q(x,y)$ are real polynomials with non-trivial dependence on $x$ and $y$, and $\epsilon$ is any positive constant. If, for a sufficiently large $n$-element set $A$ of real numbers, both $|P(A,A)|$ and $|Q(A,A)|$ are simultaneously smaller than $n^{5/4-\epsilon}$, then we shall prove that either $P(x,y) = f(u(x)+Cu(y)) \text{ and } Q(x,y) = g(u(x)+Du(y)),$ or $P(x,y) = f(u(x)u^{c}(y)) \text{ and } Q(x,y) = g(u(x)u^{d}(y)),$ where $f,g,u$ are polynomials and $C,D,c,d$ are constants. As a corollary, we obtain a strengthening of a classic result of Elekes and Rónyai in a symmetric setting of natural interest. The proof combines ideas from incidence geometry and o-minimality in model theory. This is joint work with Yifan Jing (UIUC) and Minh Chieu Tran (UIUC).

4:00 pm in 245 Altgeld Hall,Friday, March 29, 2019

#### Ants on pants

###### Agnès Beaudry   [email] (University of Colorado, Boulder)

Abstract: In this talk, I will give an introduction to manifolds and cobordism. What are manifolds? An ant living on a very large circle wouldn't know that it isn't living on the (flat) real line. In analogy, a d-manifold is a geometric object which, from an ant's perspective, looks flat like Euclidean space R^d, but which, from a bird's-eye view, can look curved or otherwise interesting, like the unit sphere in R^(d+1). What is cobordism? Think of a 2-dimensional surface that looks like a pair of empty pants. If the waist is the large circle which is the ant's universe, then the pants represent a transformation of the ant's world into a two circle universe. In analogy, a cobordism is a d+1 manifold with boundary which transforms one d-manifold into another. Two manifolds are cobordism equivalent if such a transformation exists. An interesting and difficult question is that of classifying manifolds. A raw classification in arbitrary dimensions is nearly impossible, and for this reason, mathematicians often settle for less precise answers. For example, can one classify manifolds up to cobordism equivalence? Come to my talk and find some answers to the ants on pants conundrum.

Saturday, March 30, 2019

8:00 am in Altgeld Hall,Saturday, March 30, 2019

#### Graduate Student Topology and Geometry Conference

Abstract: The Graduate Student Topology and Geometry Conference will be held March 30-31, 2019, Organizers: Hadrian Quan, Liz Tatum, Marissa Loving with faculty mentor Chris Leininger. The invited plenary speakers are Mike Hill (UCLA), Rafe Mazzeo (Stanford), and Amie Wilkinson (University of Chicago) . Visit the website for the conference schedule.

1:00 pm in Urbana,Saturday, March 30, 2019

#### Generalizing Koopman Theory to Allow for Inputs and Control

###### Kim, Hee Yeon (University of Illinois )

Sunday, March 31, 2019

8:00 am in Altgeld Hall,Sunday, March 31, 2019

#### Graduate Student Topology and Geometry Conference

Abstract: The Graduate Student Topology and Geometry Conference will be held March 30-31, 2019, Organizers: Hadrian Quan, Liz Tatum, Marissa Loving with faculty mentor Chris Leininger.

Monday, April 1, 2019

3:00 pm in 343 Altgeld Hall,Monday, April 1, 2019

###### Tsutomu Okano (UIUC Math)

Abstract: In this talk I will discuss how (higher) operads help us encode monoidal structures in (higher) categories. I will also discuss how to generalize this to parametrized settings and hope to convey the usefulness of such formalism in equivariant and motivic homotopy theories.

Tuesday, April 2, 2019

12:00 pm in 243 Altgeld Hall,Tuesday, April 2, 2019

###### Tarik Aougab (Brown University)

Abstract: The Weil-Petersson metric is a Riemannian metric on the Teichmuller space which is natural in the sense that it comes from and reflects the geometry of the hyperbolic metrics on the underlying surface. Motivated by foundational work of McMullen, Pollicott-Sharp (and later Kao) proposed an analogous metric for the moduli space of metrics on a fixed graph. We study this metric and completely characterize its completion in the case of a rose. In this talk, we’ll introduce the Weil-Petersson metric and do our best to motivate the definitions so that no advanced prior knowledge of the subject will be necessary. This represents joint work with Matt Clay and Yo’av Rieck.

1:00 pm in 345 Altgeld Hall,Tuesday, April 2, 2019

#### Max-Min theorems for weak containment, square summable homoclinic points, and completely positive entropy

###### Ben Hayes (University of Virginia)

Abstract: I will present a max-min theorem for weak containment in the context of algebraic actions (i.e. actions of a discrete group by automorphisms of a compact group). Namely, given an algebraic action of $G$ on $X$, there is a maximal, closed $G$-invariant subgroup $Y$ of $X$ so that the action of $G$ on $Y$ is weakly contained in a Bernoulli shift. This subgroup is also the minimal subgroup so that any action weakly contained in a Bernoulli shift is "$G$$X/Y-ergodic in the presence of G$$X$" (this will be defined in the talk). Time permitting, I will discussion applications. These include showing that many algebraic actions are weakly contained in a Bernoulli shift, as well as applications to complete positive entropy of algebraic actions.

1:00 pm in 347 Altgeld Hall,Tuesday, April 2, 2019

#### Direct Scattering and Small Dispersion for the Benjamin-Ono Equation with Rational Initial Data

Abstract: The Benjamin-Ono (BO) equation describes the weakly nonlinear evolution of one-dimensional interface waves in a dispersive medium. It is an integrable equation, with a known Lax pair and inverse scattering transform, that may be viewed as a prototypical problem for the study of multi-dimensional integrable equations and Riemann-Hilbert problems with a non-local jump condition. In this talk, we propose explicit formulas for the scattering data of the BO equation with a rational initial condition. For this class of initial conditions, the recovery of the scattering data can be done directly by exploiting the analyticity properties of the Lax pair solutions. Our procedure validates previous well-known formal results and provides new details concerning the leading order behavior of the scattering data in the small dispersion limit. In the small dispersion limit, we are able to derive formulas for the location and density of the eigenvalues, magnitude and phase of the reflection coefficient, and density of the phase constants.

2:00 pm in 243 Altgeld Hall,Tuesday, April 2, 2019

#### On two problems, related to additive combinatorics

###### Jozsef Balogh (Illinois Math)

Abstract: In the talk I will present two short results:

(a) Define $T=T(k)$ the minimal $t$ for which there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of the numbers $1,\dots, tn$ for all $n$, where equinumerous means that each color used the same number of times. Almost answering a question of Jungic, Licht (Fox), Mahdian, Nesetril and Radoicic, we almost determine the function $T$. It is a joint work with Linz.

(b) Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollobas in 1968. In this process, we start with initial "infected" set of edges $E(0)$, and we infect new edges according to a predetermined rule. Given a graph $H$ and a set of previously infected edges $E(t)$ subset of $E(K_n)$, we infected a non-infected edge $e$ if it completes a new copy of $H$ in $G=([n],E(t) + e)$. A question raised by Bollobas asks for the maximum time the process can run before it stabilizes. In 2015, Bollobas, Przykucki, Riordan, and Sahasrabudhe considered this problem for the most natural case where $H$ is the $r$-vertex complete graph. They answered the question for $r > 3$ and gave a lower bound for every $r \ge 5$. In their paper, they also conjectured that the maximal running time is subquadratic for every integer $r$. In this paper we disprove their conjecture for every $r$ at least 6 and we give a better lower bound for the case that $r=5$. In the proof of the case $r=5$ we use the Behrend construction. Joint result with Kronenberg, Pokrovskiy and Szabo.

4:00 pm in 243 Altgeld Hall,Tuesday, April 2, 2019

#### Generalizing Koopman Theory to Allow for Inputs and Control

###### Kim, Hee Yeon (University of Illinois, Urbana-Champaign)

Abstract: The Koopman Operator (Bernard Osgood Koopman "Hamiltonian systems and transformation in Hilbert space", PNAS 17 (1931) 315-318) has emerged in Machine Learning as a tool to reformulate nonlinear dynamics in a linear framework. I will present the paper by Proctor, Brunton, and Kutz in SIAM J.App.Dyn.Sys. (with this title) vol. 17, No. 1, 909-930.

The authors introduce the Koopman Operator with inputs and control (KIC) which generalizes Koopman's spectral theory to allow for systems with nonlinear input-output characteristics. They show how this generalization is connected to dynamic mode decompositions with control (DMDc). They demonstrate KIC on several nonlinear dynamical systems, such as the standard epidemiological SIR-model for susceptible-infectious-recovered, hence resistant subjects (e.g. measles).

5:00 pm in Ballroom, Alice Campbell Alumni Center,Tuesday, April 2, 2019

#### Department Awards Ceremony

Wednesday, April 3, 2019

3:00 pm in 243 Altgeld Hall,Wednesday, April 3, 2019

#### To Be Announced

###### Alfredo Wetzel (University of Wisconsin-Madison, Mathematics)

3:00 pm in 2 Illini Hall,Wednesday, April 3, 2019

#### Intersection Theory II

###### Yidong Chen (Illinois Physics)

Abstract: In this talk, I'll follow chapter 2 of Fulton's book and talk about divisors, pseudo-divisors, and how to intersect with divisors. As an application, I'll discuss Chern class of line bundles. With time permitting, I'll move towards the definition of Chern class of vector bundles, but will most definitely leave the actual work to the next speaker.

3:00 pm in 341 Altgeld Hall,Wednesday, April 3, 2019

#### Hurewicz' theorem (1930) on uncountable sets — a variant approach

###### Robert Kaufman (UIUC Math)

Abstract: In the theorem below, $C(K)$ is the space of continuous functions on the Cantor space $K$ and $C^*(K) \subseteq C(K)$ is the set of functions with uncountable range.

Theorem. For any analytic set $A$ in a metric space $M$, there is a continuous map $\varphi$ of $M$ into $C(K)$ such that $\varphi^{-1}(C^*(K)) = A$.

The argument uses only classical analysis; an important role is played by the notion of ultrametric space. A few minutes will be devoted to the representation of analytic sets as "projective" sets.

Thursday, April 4, 2019

11:00 am in 241 Altgeld Hall,Thursday, April 4, 2019

#### Low-lying zeros of Dirichlet L-functions

###### Kyle Pratt (Illinois Math)

Abstract: I will present work in progress with Sary Drappeau and Maksym Radziwill on low-lying zeros of Dirichlet L-functions. By way of motivation I will discuss some results on the spacings of zeros of the Riemann zeta function, and the conjectures of Katz and Sarnak relating the distribution of low-lying zeros of L-functions to eigenvalues of random matrices. I will then describe some ideas behind the proof of our theorem.

2:00 pm in 347 Altgeld Hall,Thursday, April 4, 2019

#### On the range of lattice models in high dimensions

###### Ed Perkins (University of British Columbia)

Abstract: We investigate the scaling limit of the {\em range} (the set of visited vertices) for a general class of critical lattice models, starting from a single initial particle at the origin. Conditions are given on the random sets and an associated ancestral relation" under which, conditional on longterm survival, the rescaled ranges converge weakly to the range of super-Brownian motion as random sets. These hypotheses also give precise asymptotics for the limiting behaviour of the probability of exiting a large ball, that is for the {\em extrinsic one-arm probability}. We show that these conditions are satisfied by the voter model in dimensions $d\ge2$, sufficiently spread out critical oriented percolation and critical contact processes in dimensions $d>4$, and sufficiently spread out critical lattice trees in dimensions $d>8$.

Friday, April 5, 2019

4:00 pm in 345 Altgeld Hall ,Friday, April 5, 2019

#### Generic derivations on o-minimal structures

###### Elliot Kaplan (UIUC Math)

Abstract: We study derivations $\delta$ on o-minimal fields $K$. We introduce the notion of a $T$-derivation, which is a derivation which cooperates with the 0-definable $\mathcal{C}^1$-functions on $K$. For example, if $K$ is an elementarily equivalent to the real exponential field, we require that $\delta \exp(a) = \exp(a)\delta a$ for all $a \in K$. Let $T$ be the theory of $K$ in an appropriate language $L$ and let $T^\delta$ be the $L\cup \{\delta\}$ theory stating that $\delta$ is a $T$-derivation. We show that if $T$ has quantifier elimination, then $T^\delta$ has a model completion $T^\delta_G$. The derivation in models $K$ of $T^\delta_G$ behaves "generically," it is wildly discontinuous and its kernel is a dense elementary $L$-substructure of $K$. If $T$ is the theory of real closed ordered fields, then $T^\delta_G$ is the theory of closed ordered differential fields (CODF) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that $T^\delta_G$ has $T$ as its open core and that $T^\delta_G$ is distal. This is joint work with Antongiulio Fornasiero.

4:00 pm in 145 Altgeld Hall,Friday, April 5, 2019

#### A pointless alternative to topological spaces

###### William Balderrama (UIUC)

Abstract: Fundamental to geometry and topology is the notion of a space. These are usually axiomatized as topological spaces, but there are alternative axiomatizations. In this talk, I will introduce one alternative, the locales, and describe some ways in which they can be better behaved than topological spaces.

4:00 pm in 314 Altgeld Hall,Friday, April 5, 2019

#### Integration Bee 2019 Day 1

###### Shyam Hari (UIUC Math)

Abstract: Participants will be tasked with solving 20 integrals in 30 minutes. However, these are not your ordinary Stewart's problems: these test your logic, critical thinking, mathematical intuition and knowledge! Those who perform the best will move on to day two of the Integration Bee!

Saturday, April 6, 2019

2:00 pm in 314 Altgeld Hall,Saturday, April 6, 2019

#### Integration Bee 2019 Day 2

###### Shyam Hari (UIUC Math)

Abstract: In the second day of competition, participants will be facing off head-to-head against other finalists in a three-minute problem solving race! Solving the problem correctly and fast enough will move you through the tournament!

Monday, April 8, 2019

1:00 pm in 145 Altgeld Hall,Monday, April 8, 2019

#### Uniform dimension results for the inverse images of symmetric Levy processes.

###### Hyunchul Park (SUNY New Paltz)

Abstract: In this talk, we prove the uniform Hausdorff dimension of the inverse images of a large class of symmetric Levy processes with weak scaling conditions on their characteristic exponents. Along the way we also prove an upper bound for the uniform modulus of continuity of the local times of these processes. This result extends a result of Kaufman (1985) for Brownian motions and of Song, Xiao, and Yang (2018) for stable processes. We also establish the packing dimension results as a byproduct.

3:00 pm in 343 Altgeld Hall,Monday, April 8, 2019

#### Mapping space spectral sequences

###### William Balderrama (UIUC Math)

Abstract: The classical story of obstruction theory for computing maps into a space Y involves lifting maps up the Postnikov tower of Y. In this talk, I will introduce a form of this obstruction theory for computing maps between highly structured objects in homotopy theory. Along the way, we will see why Quillen cohomologies show up in homotopy theory, take derived categories of derived categories, and take multiplicative Postnikov towers of nonconnective ring spectra.

3:00 pm in 243 Altgeld Hall,Monday, April 8, 2019

#### Shifted Poisson structures on differentiable stacks

###### Ping Xu (Pennsylvania State University)

Abstract: We will discuss shifted (+1) Poisson structures on differentiable stacks in terms of Lie groupoids. In particular, we will describe various examples and show their connection with momentum mapping theory in symplectic geometry. This is a joint work with Francesco Bonechi, Nicola Ciccoli, and Camille Laurent-Gengoux.

Tuesday, April 9, 2019

11:00 am in 345 Altgeld Hall,Tuesday, April 9, 2019

#### Classifying spectra of finite groups and chromatic homotopy theory

###### Nathan Stapleton (U Kentucky math)

Abstract: We will discuss a question about the functoriality of certain evaluation maps for classifying spectra of finite groups that arose when thinking about questions related to chromatic homotopy theory. I will describe a solution to this problem found in joint work with Reeh, Schlank.

12:00 pm in 243 Altgeld Hall,Tuesday, April 9, 2019

#### Geometry Group Theory in Music AI

###### Haizi Yu (University of Illinois)

Abstract: Is it possible to invent an AI to learn important music concepts directly from sheet music? Is it possible to do this in a human-interpretable form that resembles known music theory and also suggests new theory? We apply our generally developed automatic concept learning model to the domain of music, so as to tackle the above questions and the like. Starting from a connection between existing music concepts and their group-theoretic interpretations, we propose a formal representation of music objects as well as their abstractions and probabilistic patterns. This proposed representation not only reveals internal music structures as mathematical symmetries, but more importantly, are also operational in computational models. As a result, this further yields a learning algorithm that couples knowledge from geometric group theory and statistical inference to automatically discover music concepts without human intervention. Lastly, we briefly demonstrate an ongoing project, called MUS-ROVER, which builds a real web application that delivers to people automatically discovered music rules and concepts, teaching us music composition in a designated style.

1:00 pm in 345 Altgeld Hall,Tuesday, April 9, 2019

#### Multiplication of weak equivalence classes

###### Anton Bernshteyn (Carnegie Mellon)

Abstract: The relations of weak containment and weak equivalence were introduced by Kechris in order to provide a convenient framework for describing global properties of p.m.p. actions of countable groups. Weak equivalence is a rather coarse relation, which makes it relatively well-behaved; in particular, the set of all weak equivalence classes of p.m.p. actions of a given countable group $\Gamma$ carries a natural compact metrizable topology. Nevertheless, a lot of useful information about an action (such as its cost, type, etc.) can be recovered from its weak equivalence class. In addition to the topology, the space of weak equivalence classes is equipped with a multiplication operation, induced by taking products of actions, and it is natural to wonder whether this multiplication operation is continuous. The answer is positive for amenable groups, as was shown by Burton, Kechris, and Tamuz. In this talk, we will explore what happens in the nonamenable case. Number theory will make an appearance.

1:00 pm in 347 Altgeld Hall,Tuesday, April 9, 2019

#### Convexity of Whitham's wave of extreme form

###### Bruno Vergara (ICMAT, Spain)

Abstract: In this talk I will discuss a conjecture of Ehrnström and Wahlén concerning travelling wave solutions of greatest height to Whitham's non-local model of water waves. We will see that there exists a cusped periodic solution whose profile is convex between consecutive peaks of $C^{1/2}$-regularity. The talk is based on joint work with A. Enciso and J. Gómez-Serrano.

2:00 pm in 345 Altgeld Hall,Tuesday, April 9, 2019

#### Quantitative inequalities for the expected lifetime of the Brownian motion

###### Daesung Kim (Purdue University)

Abstract: The isoperimetric-type inequality for the expected lifetime of the Brownian motion state that the $L^p$ norm of the expected lifetime in a region is maximized when the region is a ball with the same volume. In particular, if $p=1$, it is called the Saint-Venant inequality and has a close relation to the classical Faber—Krahn inequality for the first eigenvalue. In this talk, we prove a quantitative improvement of the inequalities, which explains how a region is close to being a ball when equality almost holds in these inequalities. We also discuss some related open problems.

2:00 pm in 243 Altgeld Hall,Tuesday, April 9, 2019

#### Equitable colorings of infinite graphs

###### Anton Bernshteyn (Carnegie Mellon Math)

Abstract: A proper $k$-coloring of a finite graph $G$ is called equitable if every two color classes differ in size at most by one. In particular, if $G$ has $n$ vertices and $k$ divides $n$, then in an equitable $k$-coloring of $G$ every color class has size exactly $n/k$. There is a natural way to extend this definition to infinite graphs on probability spaces. Namely, if $G$ is a graph whose vertex set $V(G)$ is a probability space, then a proper $k$-coloring of $G$ is equitable when every color class has measure $1/k$. In this talk I will discuss extensions of some classical results about equitable colorings to this setting, including an infinite version of the Hajnal-Szemerédi theorem on equitable $k$-colorings for $k \geq \Delta(G) + 1$, and an analog of the Kostochka-Nakprasit theorem on equitable $\Delta$-colorings of graphs with small average degree. This is joint work with Clinton Conley.

3:00 pm in 243 Altgeld Hall,Tuesday, April 9, 2019

#### Quantization of algebraic exact Lagrangians in cotangent bundles

###### Christopher Dodd (UIUC Math)

Abstract: Exact Lagrangians play an important role in symplectic topology; in algebraic geometry they seem to be almost unstudied. In this talk I’ll explain some recent results about their structure and in particular I’ll show that, in the affine case, they admit certain canonical noncommutative deformations. Time permitting I’ll explain how this implies the vanishing of certain invariants in their de Rham cohomology.

4:00 pm in 314 Altgeld Hall,Tuesday, April 9, 2019

#### Recent progress on existence of minimal surfaces

###### André Neves (University of Chicago)

Abstract: The Tondeur Memorial Lectures will be given by Andre Neves (University of Chicago), April 9-11, 2019. Following this lecture, a reception will be held in 239 Altgeld Hall.

A long standing problem in geometry, conjectured by Yau in 1982, is that any any $3$-manifold admits an infinite number of distinct minimal surfaces. The analogous problem for geodesics on surfaces led to the discovery of deep interactions between dynamics, topology, and analysis. The last couple of years brought dramatic developments to Yau’s conjecture, which has now been settled due to the work of Marques-Neves and Song. In the first talk I will survey the history of the problem and the several contributions made. In the second talk I will talk about the Weyl law for the volume spectrum (Marques-Neves-Liokumovich) and how it can be used to prove denseness and equidistribution of minimal surfaces in the generic case (Irie-Marques-Neves and Marques-Neves-Song). In the third talk I will survey the recent breakthroughs due to Song, Zhou, and Mantoulidis-Chodosh.

Bio Note: André Neves is a leading figure in geometric analysis with important contributions ranging from the Yamabe problem to geometric flows. Jointly with Fernando Marques, he transformed the field by introducing new ideas and techniques that led to the solution of several open problems which were previously out of reach. Together or with coauthors, they solved the Willmore conjecture, the Freedman-He-Wang conjecture in knot theory and Yau’s conjecture on the existence of minimal surfaces in the generic case.

Neves received his PhD from Stanford University in 2005 under the supervision of Richard Schoen. He was a postdoctoral fellow and assistant professor at Princeton University, before joining the Imperial College of London in 2011, where he became a full professor. He joined the faculty of the University of Chicago in 2016. Among his many awards and recognitions, Neves was awarded the Philip Leverhulme Prize in 2012, the LMS Whitehead Prize in 2013, he was invited speaker at ICM in Seoul in 2014, received a New Horizons in Mathematics Prize in 2015, and the 2016 Oswald Veblen Prize in Geometry. In 2018, he received a Simons Investigator Award.

Wednesday, April 10, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, April 10, 2019

#### Coloring Borel graphs equitably

###### Anton Bernshteyn (Carnegie Mellon)

Abstract: In this talk I will describe some of the main ideals and tools behind the proofs of the results surveyed in my talk in the Combinatorics and Graph Theory Seminar yesterday (based on joint work with Clinton Conley).

3:00 pm in 2 Illini Hall,Wednesday, April 10, 2019

#### Intersection Theory III - Chern classes of vector bundles

Abstract: In this talk, based on chapter 3 of Fulton's "Intersection Theory", I will introduce Segre classes and Chern classes, and outline some of their basic properties. I will also discuss a few interesting examples and special cases.

3:00 pm in 243 Altgeld Hall,Wednesday, April 10, 2019

#### To Be Announced

###### Danielle Sass (University of Illinois, Statistics)

4:00 pm in 245 Altgeld Hall,Wednesday, April 10, 2019

#### Recent progress on existence of minimal surfaces

###### André Neves (University of Chicago)

Abstract: A long standing problem in geometry, conjectured by Yau in 1982, is that any any $3$-manifold admits an infinite number of distinct minimal surfaces. The analogous problem for geodesics on surfaces led to the discovery of deep interactions between dynamics, topology, and analysis. The last couple of years brought dramatic developments to Yau’s conjecture, which has now been settled due to the work of Marques-Neves and Song. In the first talk I will survey the history of the problem and the several contributions made. In the second talk I will talk about the Weyl law for the volume spectrum (Marques-Neves-Liokumovich) and how it can be used to prove denseness and equidistribution of minimal surfaces in the generic case (Irie-Marques-Neves and Marques-Neves-Song). In the third talk I will survey the recent breakthroughs due to Song, Zhou, and Mantoulidis-Chodosh.

Bio Note: André Neves is a leading figure in geometric analysis with important contributions ranging from the Yamabe problem to geometric flows. Jointly with Fernando Marques, he transformed the field by introducing new ideas and techniques that led to the solution of several open problems which were previously out of reach. Together or with coauthors, they solved the Willmore conjecture, the Freedman-He-Wang conjecture in knot theory and Yau’s conjecture on the existence of minimal surfaces in the generic case.

Neves received his PhD from Stanford University in 2005 under the supervision of Richard Schoen. He was a postdoctoral fellow and assistant professor at Princeton University, before joining the Imperial College of London in 2011, where he became a full professor. He joined the faculty of the University of Chicago in 2016. Among his many awards and recognitions, Neves was awarded the Philip Leverhulme Prize in 2012, the LMS Whitehead Prize in 2013, he was invited speaker at ICM in Seoul in 2014, received a New Horizons in Mathematics Prize in 2015, and the 2016 Oswald Veblen Prize in Geometry. In 2018, he received a Simons Investigator Award.

Thursday, April 11, 2019

11:00 am in 241 Altgeld Hall,Thursday, April 11, 2019

#### Vanishing of Hyperelliptic L-functions at the Central Point

###### Wanlin Li (Wisconsin Math)

Abstract: We study the number of quadratic Dirichlet L-functions over the rational function field which vanish at the central point s=1/2. In the first half of my talk, I will give a lower bound on the number of such characters through a geometric interpretation. This is in contrast with the situation over the rational numbers, where a conjecture of Chowla predicts there should be no such L-functions. In the second half of the talk, I will discuss joint work with Ellenberg and Shusterman proving as the size of the constant field grows to infinity, the set of L-functions vanishing at the central point has 0 density.

12:30 pm in 464 Loomis,Thursday, April 11, 2019

#### Effective field theory and effective response away from equilibrium

###### Paolo Glorioso (University of Chicago)

Abstract: In the first part of this talk I will describe how the formalism of non-equilibrium effective field theory (EFT) provides a field-theoretical description of the low-energy behavior of systems in local thermal equilibrium. I will then show how magnetohydrodynamics can be incorporated in this formalism using generalized global symmetries. In the second part of the talk I will discuss response for Floquet systems, which do not possess a notion of equilibrium, and for which we lack of an effective theory formulation. I will show how this can be remedied by applying the approach of non-equilibrium EFT to describe topological response of such systems.

2:00 pm in 241 Altgeld Hall,Thursday, April 11, 2019

#### Conversations on the exceptional character

Abstract: We will spend the last few weeks of the semester discussing Landau-Siegel zeros. In particular, we will be discussing Henryk Iwaniec's survey article "Conversations on the exceptional character."

2:00 pm in 347 Altgeld Hall,Thursday, April 11, 2019

#### Local Limit Theorem

###### Qiang Wu (UIUC Math)

Abstract: This talk is an introduction to some classical CLT variants, specifically on local limit theorem (LLT). The proof of classical LLT for lattice and non-lattice distribution will be discussed using the characteristic approach. Other various generalizations of LLT will be pointed out. Finally, a concise combinatorial approach for LLT of simple random walk will be sketched. Time permits, I will talk about the generalized Berry-Esseen Inequality.

4:00 pm in 245 Altgeld Hall,Thursday, April 11, 2019

#### Recent progress on existence of minimal surfaces

###### André Neves (University of Chicago)

Abstract: A long standing problem in geometry, conjectured by Yau in 1982, is that any any $3$-manifold admits an infinite number of distinct minimal surfaces. The analogous problem for geodesics on surfaces led to the discovery of deep interactions between dynamics, topology, and analysis. The last couple of years brought dramatic developments to Yau’s conjecture, which has now been settled due to the work of Marques-Neves and Song. In the first talk I will survey the history of the problem and the several contributions made. In the second talk I will talk about the Weyl law for the volume spectrum (Marques-Neves-Liokumovich) and how it can be used to prove denseness and equidistribution of minimal surfaces in the generic case (Irie-Marques-Neves and Marques-Neves-Song). In the third talk I will survey the recent breakthroughs due to Song, Zhou, and Mantoulidis-Chodosh.

Bio Note: André Neves is a leading figure in geometric analysis with important contributions ranging from the Yamabe problem to geometric flows. Jointly with Fernando Marques, he transformed the field by introducing new ideas and techniques that led to the solution of several open problems which were previously out of reach. Together or with coauthors, they solved the Willmore conjecture, the Freedman-He-Wang conjecture in knot theory and Yau’s conjecture on the existence of minimal surfaces in the generic case.

Neves received his PhD from Stanford University in 2005 under the supervision of Richard Schoen. He was a postdoctoral fellow and assistant professor at Princeton University, before joining the Imperial College of London in 2011, where he became a full professor. He joined the faculty of the University of Chicago in 2016. Among his many awards and recognitions, Neves was awarded the Philip Leverhulme Prize in 2012, the LMS Whitehead Prize in 2013, he was invited speaker at ICM in Seoul in 2014, received a New Horizons in Mathematics Prize in 2015, and the 2016 Oswald Veblen Prize in Geometry. In 2018, he received a Simons Investigator Award.

5:00 pm in TBA,Thursday, April 11, 2019

#### To Be Announced

Friday, April 12, 2019

4:00 pm in 145 Altgeld Hall,Friday, April 12, 2019

#### What is a Higgs bundle?

###### Matej Penciak (UIUC)

Abstract: In this talk I will introduce and try to motivate Higgs bundles as objects that naturally arise in algebra and geometry.

4:00 pm in 345 Altgeld Hall ,Friday, April 12, 2019

#### Ultraproducts as a tool in the model theory of metric structures

###### Ward Henson (UIUC)

Abstract: L is a signature of continuous first order logic for metric structures and we have a class C of L-structures which we want to investigate from the point of view of model theory. In general, this involves letting T be the L-theory of C, and working to understand the models of T as fully as possible. This means not only knowing which L-structures are models of T, but also understanding the definable predicates and (especially important) the definable sets in models of T. (A valuable byproduct might be an explicit axiomatization of T.) In this talk we will lay out how understanding ultraproducts of members of C can be an important practical tool for understanding the full class of models of T. As much as time permits, we will discuss examples that have been successfully treated in this way, including some new ones, focusing on Banach spaces and Banach lattices. (Most of this work on examples is part of a collaboration with Yves Raynaud.)

4:00 pm in 241 Altgeld Hall,Friday, April 12, 2019

#### Beatty Sequences

###### Xiaomin Li (UIUC Math)

Abstract: A Beatty sequence is a sequence of the form [a*n], where a is an irrational number and the bracket denotes the floor function. A remarkable result, called Beatty's Theorem, says that if a and b are irrational numbers such that 1/a+1/b=1, then the associated Beatty sequences "partition" the natural numbers. That is, every natural number belongs to exactly one of these two sequences. It is known that Beatty's Theorem does not extend directly to partitions into three or more sets, and finding appropriate analogs of Beatty's Theorem for such partitions is an interesting, and wide open, problem, which has applications to optimal scheduling questions. The goal of this project is to explore different constructions of partitions of integers into perturbed Beatty sequences and possible applications to optimal scheduling algorithms.

Monday, April 15, 2019

3:00 pm in 343 Altgeld Hall,Monday, April 15, 2019

#### Group Theory for Homotopy Theorists

###### Brian Shin (UIUC Math)

Abstract: In this expository talk, we'll introduce a model structure on the category of groups and demonstrate how to effectively study groups using this model. This model has the technical advantage of avoiding the overly abstract definition of a group via sets with binary operation. It also allows for clean definitions of colimits and free objects. If time permits, we'll discuss monoidal structures for a certain localization of this model structure. This is based on a short article by Krause-Nikolaus.

5:00 pm in 241 Altgeld Hall,Monday, April 15, 2019

#### Complete Logarithmic Sobolev Inequalities (CSLI) and Ricci Curvature

###### Haojian Li (UIUC)

Abstract: First we would continue Eddie's talk and start with how the connection on a G-bundle induces a connection on the associated bundle naturally. Then a brief introduction about the quantum information will be included. We would focus on formulating the CLSI problem today and show that the CLSI constants depend on the lower bound of Ricci curvature. If time permits, we would also apply our machinery to Hormander system.

Tuesday, April 16, 2019

11:00 am in 345 Altgeld Hall,Tuesday, April 16, 2019

#### Iterated K-theory of the integers and higher Lichtenbaum-Quillen conjectures

###### Gabe Angelini-Knoll (Michigan State University)

Abstract: The Hurewicz image of the alpha family in the algebraic K-theory of the integers is know to correspond to special values of the Riemann zeta function, by work of Adams and Quillen. Lichtenbaum and Quillen conjectured that, more generally, there should be a relationship between special values of Dedekind zeta functions and algebraic K-theory. These conjectures have now largely been proven by work of Voevodsky and Rost. The red-shift conjectures of Ausoni-Rognes generalize the Lichtenbaum-Quillen conjecture to higher chromatic heights in a precise sense. In that same spirit, I conjecture that the n-th Greek letter family is detected in the Hurewicz image of the n-th iteration of algebraic K-theory of the integers. In my talk, I will sketch a proof of this conjecture in the case n=2 using the theory of trace methods. Specifically, I prove that the beta family is detected in the Hurewicz image of iterated algebraic K-theory of the integers. This is a higher chromatic height analogue of the result of Adams and Quillen. Consequently, by work of Behrens, Laures, and Larson iterated algebraic K-theory of the integers detects explicit information about certain modular forms.

1:00 pm in 345 Altgeld Hall,Tuesday, April 16, 2019

#### Positive model theory and sober spaces

###### Levon Haykazyan (University of Waterloo)

Abstract: I will talk about positive model theory (also known as coherent logic) where formulas are not closed under negation. This setting is in fact more general that full first-order logic, since negation can be expressed by changing the language. The result is that we can have as much negation as necessary, however no extra negation is forced by the framework.
We can associate to a positive theory a natural spaces of types, which will no longer be Hausdorff, but (quasi-)compact and sober. I will show that these spaces play the role of the Stone spaces in the full first-order logic. In particular I will show how classical results (due to Vaught) connecting the structure of countable models to Stone spaces carry over to the positive setting, provided we find the appropriate formulations of topological properties for non-Hausdorff spaces.

2:00 pm in 345 Altgeld Hall,Tuesday, April 16, 2019

#### Large deviations for quasilinear parabolic stochastic partial differential equations

###### Rangrang Zhang (Beijing Institute of Technology and University of Tennessee)

Abstract: In this talk I will present some recent results on large deviations for quasilinear parabolic stochastic partial differential equations. More precisely, I will talk about Freidlin-Wentzell type large deviations for quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are not necessarily locally monotone. Our proof is based on the weak convergence approach.

2:00 pm in 243 Altgeld Hall,Tuesday, April 16, 2019

#### Monochromatic connected matchings, paths and cycles in 2-edge-colored multipartite graphs

###### Xujun Liu (Illinois Math)

Abstract: We solve four similar problems: For every fixed $s$ and large $n$, we describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph $K_{n_1,\ldots,n_s}$ there exists a monochromatic
(i) cycle $C_{2n}$ with $2n$ vertices,
(ii) cycle $C_{\geq 2n}$ with at least $2n$ vertices,
(iii) path $P_{2n}$ with $2n$ vertices, and
(iv) path $P_{2n+1}$ with $2n+1$ vertices.

This implies a generalization of the conjecture by Gyárfás, Ruszinkó, Sárközy and Szemerédi that for every $2$-edge-coloring of the complete $3$-partite graph $K_{n,n,n}$ there is a monochromatic path $P_{2n+1}$. An important tool is our recent stability theorem on monochromatic connected matchings (A matching $M$ in $G$ is connected if all the edges of $M$ are in the same component of $G$). We will also talk about exact Ramsey-type bounds on the sizes of monochromatic connected matchings in $2$-colored multipartite graphs. Joint work with József Balogh, Alexandr Kostochka and Mikhail Lavrov.

2:00 pm in 347 Altgeld Hall,Tuesday, April 16, 2019

#### Testing families of analytic discs

###### Luca Baracco (University of Padova, Italy)

Abstract: It is a well-known fact in the theory of several complex variables that a function is holomorphic if and only if it is holomorphic in each variable separately. This result goes back to Hartogs. It is natural to consider a boundary version of Hartogs’ theorem. The general problem is to take a boundary function and ask if holomorphic extensions on some families of complex curves are enough to guarantee an extension which is holomorphic in all variables simultaneously. We will talk about the known results on the subject and show some new results obtained in collaboration with M. Fassina and S. Pinton for the special case of the unit ball in ${\mathbb C}^n$.

3:00 pm in 245 Altgeld Hall,Tuesday, April 16, 2019

#### An Integrated Approach to Measuring Asset and Liability Risks in Financial Institutions

###### George Zanjani (Professor of Finance and the Frank Park Samford Chair of Insurance, University of Alabama)

Abstract: Risk measurement models for financial institutions typically focus on the net portfolio position and thus ignore distinctions between 1) assets and liabilities and 2) uncollateralized and collateralized liabilities. However, these distinctions are economically important. Liability risks affect the total amount of claims on the institution, while asset risks affect the amount available for claimants. Collateralization also affects the amounts recovered by different classes of claimants. We analyze a model of a financial institution with risky assets and liabilities, with potentially varying levels of collateralization across liabilities, showing that correct economic risk capital allocation requires complete segregation of asset, uncollateralized liability, and collateralized liability risks, with different risk measures for each. Our numerical analyses suggest that the conventional approach frequently yields over-investment in risky assets.

Bio: George Zanjani is Professor of Finance and the Frank Park Samford Chair of Insurance at the University of Alabama. Previously, he served as the inaugural holder of the AAMGA Distinguished Chair in Risk Management and Insurance and an associate professor in the RMI Department of Georgia State University. Prior to his career in academia, he served as an economist at the Federal Reserve Bank of New York (2000–2008) specializing in policy work relating to insurance issues in the broader financial system. During his tenure at the Bank, he served on working groups formed by the Committee on the Global Financial System and the Presidential Working Group on Financial Markets. He also worked as an actuary at Fireman’s Fund Insurance Companies (1990–1994), focusing on commercial insurance pricing and heading the firm’s workers’ compensation actuarial unit in 1994.

Dr. Zanjani's published or forthcoming work includes insurance papers in the American Economic Review, Insurance: Mathematics and Economics, the Journal of Financial Economics, the Journal of Public Economics, the Journal of Risk and Insurance, Management Science, and the North American Actuarial Journal. He has served on working groups formed by the Committee on the Global Financial System (on global savings and asset allocation) and the Presidential Working Group on Financial Markets (terrorism insurance).

Dr. Zanjani is an Associate of the Casualty Actuarial Society. He earned his A.B./B.S. in Economics and Biology from Stanford University and holds a Ph.D. in Economics from the University of Chicago. He served as the President of both the American Risk and Insurance Association and the Risk Theory Society.

3:00 pm in 243 Altgeld Hall,Tuesday, April 16, 2019

#### Stokes decompositions and wild monodromy

###### Philip Boalch (Orsay)

Abstract: Just like a Hodge structure can be described equivalently in terms of the Hodge filtration or the Hodge decomposition, a Stokes structure has several equivalent descriptions. The best known are the Stokes filtrations and the Stokes local systems (or wild monodromy representations). In this talk I will explain how to formalise the notion of {\em Stokes decompositions}, to intermediate between them. This is part of an attempt (the Lax project) to understand the bestiary of complete hyperkahler manifolds that occur as moduli spaces of algebraic Higgs bundles on the affine line.

4:00 pm in 243 Altgeld Hall,Tuesday, April 16, 2019

#### Visualizing nonlinear dynamical systems like SIR

###### George K Francis   [email] (University of Illinois at Urbana–Champaign)

Abstract: There is no new presentation this week. But ...

.. for those of you who are interested in programming real-time interactive computer animations of non-linear dynamical systems, like the SIR system we saw last week in Heejeon's seminar on Koopman's theory, I will be there to introduce you to the issues and and problems involved. Recall that the SIR models the epidemiological progress of three populations: Susceptible, Infected, Recovered from the disease (thinks of measles or mumps).

In the first of (possibly) two such workshops I will treat the "continuous" case, which involves some (elementary) integration of 3D differential systems and their steady-states (attractors). In the (tentative) second workshop I will treat the "discrete" case, animating cellular automata, since both are relevant to the SIR model.

5:00 pm in TBA,Tuesday, April 16, 2019

Wednesday, April 17, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, April 17, 2019

#### Introduction to quasi-Polish spaces

###### Ruiyuan (Ronnie) Chen (UIUC)

Abstract: We give an introduction to de Brecht's quasi-Polish spaces, a possibly non-Hausdorff generalization of Polish spaces sharing most of their descriptive set-theoretic properties while enjoying some additional and highly useful closure properties.

3:00 pm in 2 Illini Hall,Wednesday, April 17, 2019

#### Intersection Theory IV

###### Jin Hyung To (Illinois Math)

Abstract: We study Section 4. We construct the Segre class of a closed subscheme which is a cycle class of the subscheme.

4:00 pm in 245 Altgeld Hall,Wednesday, April 17, 2019

#### From Graph Laplacian to the Stability of Coupled Oscillator Networks

###### Lan Wang (Illinois Math)

Abstract: There is a large amount of applied problems that can be posed as dynamical systems on a coupled oscillator network. Frequently these problems involve computing the inertia of a graph Laplacian. In this talk we will start with an overview of the properties of the Laplacian matrix and then explore how it functions in the study of the stability of fixed points of dynamical systems. Particularly, we will discuss the Kuramoto model, a classic and popular model for describing the dynamics of a large population of coupled oscillators. We will first deliberate the stability of the phase-locked solutions of Kuramoto model on single-layer networks, and then extend it to multi-layer networks by examining the Supra-Laplacian matrix.

Thursday, April 18, 2019

12:00 pm in 243 Altgeld Hall,Thursday, April 18, 2019

#### Immersions and Laminations on Free Groups

###### Jean-Pierre Mutanguha (Arkansas Math)

Abstract: Using pullbacks, we proved that mapping tori of graph immersions have word-hyperbolic fundamental groups if and only if they have no Baumslag-Solitar subgroups. We will then use laminations to describe an efficient algorithm that determines whether such groups are word-hyperbolic.

2:00 pm in 347 Altgeld Hall,Thursday, April 18, 2019

#### Local Limit Theorem (Part 2)

###### Qiang Wu (UIUC Math)

Abstract: This talk the second part of an introduction to some classical CLT variants, specifically on local limit theorem (LLT). The proof of classical LLT for lattice and non-lattice distribution will be discussed using the characteristic approach. Other various generalizations of LLT will be pointed out. Finally, a concise combinatorial approach for LLT of simple random walk will be sketched. Time permits, I will talk about the generalized Berry-Esseen Inequality.

4:00 pm in 245 Altgeld Hall,Thursday, April 18, 2019

#### The many aspects of Schubert polynomials

###### Karola Mészáros (Cornell University)

Abstract: Schubert polynomials, introduced by Lascoux and Schützenberger in 1982, represent cohomology classes of Schubert cycles in flag varieties. While there are a number of combinatorial formulas for Schubert polynomials, their supports have only recently been established and the values of their coefficients are not well understood. We show that the Newton polytope of a Schubert polynomial is a generalized permutahedron and explain how to obtain certain Schubert polynomials as projections of integer point transforms of polytopes. The latter generalizes the well-known relationship between Schur functions and Gelfand-Tsetlin polytopes. We will then turn to the study of the coefficients of Schubert polynomials and show that Schubert polynomials with all coefficients at most $k$, for any positive integer $k$, are closed under pattern containment. We also characterize zero-one Schubert polynomials by a list of twelve avoided patterns. This talk is based on joint works with Alex Fink, Ricky Liu and Avery St. Dizier.

Friday, April 19, 2019

2:00 pm in 141 Altgeld Hall,Friday, April 19, 2019

#### Universality in Operator Spaces

###### Mary Angelica Gramcko-Tursi (Illinois Math)

Abstract: Given a class $\mathcal{C}$ of spaces, When does there exist a space $\mathcal{U}$ that is injectively or projectively universal for $\mathcal{C}$ under the appropriate operation-preserving mappings?  Furthermore, when is $\mathcal{U}$ in $\mathcal{C}$ ?  The question has been answered under certain conditions using tools both in analysis and logic. We will look at both classical and recent results, as well as some of the techniques used to arrive at them. If time permits, we will end with some open questions.

4:00 pm in 345 Altgeld Hall ,Friday, April 19, 2019

#### Cancelled

###### (UIUC Math)

4:00 pm in 241 Altgeld Hall,Friday, April 19, 2019

#### Introduction to Generating Functions

###### Longzheng Chen (UIUC Math)

4:00 pm in 145 Altgeld Hall,Friday, April 19, 2019

#### Complex structures on real vector bundles

###### Abhra Abir Kundu (UIUC)

Abstract: In this talk, I will provide an interpretation of the question "Does a given real vector bundle admit a complex structure?" and offer an approach to understanding this question.

Monday, April 22, 2019

3:00 pm in 343 Altgeld Hall,Monday, April 22, 2019

#### Complex structures on Real vector bundles

###### Abhra Abir Kundu (UIUC Math)

Abstract: In this talk, I will state the first and the second obstruction to having a stable complex structure on a real vector bundle. I will then show how one can go from stable complex structure to complex structure. And, if time permits, I will try to sketch how the second obstruction can be expressed as a secondary cohomology operation.

5:00 pm in Altgeld Hall,Monday, April 22, 2019

#### The complete logarithmic Sobolev inequality and Ricci curvature.

###### Haojian Li (UIUC)

Abstract: Today we are going to prove the main theorem and show how the lower bound of Ricci curvature get involved with algebra.

Tuesday, April 23, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, April 23, 2019

#### On Hardy-Rellich-type inequalities

###### Fritz Gesztesy (Baylor University)

Abstract: We will illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisely, using this factorization method, we will derive a general inequality and demonstrate how particular choices of the parameters contained in this inequality yield well-known inequalities, such as the classical Hardy and Rellich inequalities, as special cases. Actually, other special cases yield additional and apparently less well-known inequalities. We will indicate that our method is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order operators. If time permits, we might illustrate a very recent new and most elementary proof in the one-dimensional context. This talk will be accessible to students. This is based on joint work with Lance Littlejohn, Isaac Michael, and Michael Pang.

1:00 pm in 345 Altgeld Hall ,Tuesday, April 23, 2019

#### Expansions of the real field which does not introduce new smooth functions

###### Alex Savatovsky (Universität Konstanz)

Abstract: We will give some conditions under which an expansion of the real field does not define new smooth functions. We will give a very rough sketch of the proof and discuss generalizations.

2:00 pm in 243 Altgeld Hall,Tuesday, April 23, 2019

#### Partitions of hypergraphs under variable degeneracy constraints

###### Michael Stiebitz (TU Ilmenau)

Abstract: We use the concept of variable degeneracy of a hypergraph in order to unify the seemingly remote problems of determining the point partition numbers and the list chromatic number of hypergraphs. Our hypergraphs may have multiple edges, but no loops. Given a hypergraph $G$ and a sequence $f = (f_1, f_2, \dots , f_p)$ of $p \ge 1$ vertex functions $f_i : V(G) → \mathbb N_0$ such that $f_1(v) + f_2(v) + · · · + f_p(v) \ge d_G(v)$ for all $v \in V(G)$, we want to find a sequence $(G_1, G_2, \dots , G_p)$ of vertex disjoint induced subhypergraphs containing all vertices of $G$ such that each hypergraph G_i is strictly $f_i$-degenerate, that is, for every non-empty subhypergraph $G' \subseteq G_i$ there is a vertex $v \in V (G')$ such that $d_{G'}(v) < f_i(v)$. The main result says that such a sequence of hypergraphs exists if and only if $(G, f)$ is not a so-called hard pair. Hard pairs form a recursively defined family of configurations, obtained from three basic types of configurations by the operation of merging a vertex. For simple graphs this result was obtained by O. Borodin, A. V. Kostochka, and B. Toft in 2000. As a simple consequence of our result we obtain a Brooks-type result for the list chromatic number of digraphs due to A. Harautyunyan and B. Mohar. In a digraph coloring the aim is to color the vertices of a directed graph $D$ such that each color class induces an acyclic digraph of $D$, that is, a directed graph not containing any directed cycle. This coloring concept was introduced by V. Neumann-Lara in the 1980s.

3:00 pm in 243 Altgeld Hall,Tuesday, April 23, 2019

#### Virtual Euler characteristics of Quot scheme of surfaces

###### Rahul Pandharipande (ETH Zurich)

Abstract: Let S be a nonsingular projective surface. Quot schemes of quotients on S with supports of dimensions 0 and 1 always have 2-term obstruction theories (and therefore also have natural virtual fundamental classes). I will explain what we know about the virtual Euler characteristics in this theory: theorems, conjectures, and a lot of examples. Joint work with Dragos Oprea.

4:00 pm in 245 Altgeld Hall,Tuesday, April 23, 2019

#### The challenge of modeling dryland vegetation pattern formation using ideas from dynamical systems

###### Mary Silber (University of Chicago)

Abstract: A beautiful example of spontaneous pattern formation appears in the distribution of vegetation in some dry-land environments. Examples from Africa, Australia and the Americas reveal that vegetation, at a community scale, may spontaneously form into stripe-like bands, alternating with striking regularity with bands of bare soil, in response to aridity stress. A typical length scale for such patterns is 100 m; they are readily surveyed by modern satellites (and explored from your armchair in Google maps). These ecosystems represent some of Earth’s most vulnerable under threats to desertification, and some ecologists have suggested that the patterns, so easily monitored by satellites, may have potential as early warning signs of ecosystem collapse. I will describe efforts based in simple mathematical models, inspired by decades of physics research on pattern formation, to understand the morphology of the patterns, focusing particularly on topographic influences. I will take a critical look at the role of mathematical models in developing potential remote probes of these ecosystems. How does mathematical modeling influence what we see? Does it suggest what we should monitor? Could it lead us astray?

Wednesday, April 24, 2019

3:00 pm in 2 Illini Hall,Wednesday, April 24, 2019

#### Intersection Theory V-Intersection Products

###### Sungwoo Nam (Illinois Math)

Abstract: In this talk, we will see the important construction of deformation to the normal cone, which is an analog of the tubular neighborhood theorem in algebraic geometry. Using this, we will define intersection product with a regular codimension d subvariety, generalizing intersection with a divisor introduced in the second talk. Time permitting, we will see how to understand the number 3264 from the intersection theory point of view.

3:00 pm in 243 Altgeld Hall,Wednesday, April 24, 2019

#### To Be Announced

###### Mary Silber (University of Chicago, Statistics)

4:00 pm in 245 Altgeld Hall,Wednesday, April 24, 2019

#### Disable the Label: A Dialogue on Ableism

###### TBA (Office of Inclusion & Intercultural Relations)

Abstract: Disable the Label: A Dialogue on Ableism examines issues related to disability and ableism, including an introduction to accommodations, how our physical, social, and cultural environment defines disability, and how to be an ally to people with disabilities. Participants will leave the workshop with resources to continue the conversation about disability justice.

Thursday, April 25, 2019

11:00 am in 241 Altgeld Hall,Thursday, April 25, 2019

#### Local models for potentially crystalline deformation rings and the Breuil-Mézard conjecture

###### Stefano Morra (Paris 8)

Abstract: Available at https://faculty.math.illinois.edu/~pballen/stefano-morra-abstract.pdf

12:00 pm in 243 Altgeld Hall,Thursday, April 25, 2019

#### Mirzakhani's curve counting

###### Viveka Erlandsson (U Bristol)

Abstract: Mirzakhani proved two theorems about the asymptotic growth of the number of curves in a mapping class group orbit on a surface: one for simple curves and another for general curves. In this talk I will explain how to derive her second theorem from the one about simple curves. Time permitting, I will explain why similar methods can be used to also give a proof for the theorem about simple curves, hence giving a new (and very different) proof of both theorems.

2:00 pm in 243 Altgeld Hall,Thursday, April 25, 2019

#### Classification of irreversible and reversible operator algebras

Abstract: C*-algebras have been studied quite extensively in the literature, especially in an attempt to classify them using K-theory. One canonical example is classification of Cuntz-Krieger algebras of a directed graph where K-theory was shown to coincide with Bowen-Franks groups of the subshift associated to the graph. On the other hand, non-self-adjoint operator algebras have been used to encode one-sided processes such as continuous maps on a compact space, stochastic matrices and graphs in their own right. In this talk we will survey results from both irreversible and reversible classification, and uncover a beautiful hierarchy of classification results for irreversible and reversible operator algebras.

2:00 pm in 347 Altgeld Hall,Thursday, April 25, 2019

#### Coupling and its applications

###### Peixue Wu (UIUC Math)

Abstract: I will define what is coupling. The beginning example is the transport problem, which leads to the concepts of optimal coupling and probability distance. We will also talk about applications of coupling to study ergodicity, gradient estimate and Harnack's inequality for Markov processes.

4:00 pm in 245 Altgeld Hall,Thursday, April 25, 2019

#### Spring Department Faculty Meeting

Abstract: The Spring Department Faculty Meeting will be held at 4 p.m. in 245 Altgeld Hall, followed by a reception in 239 Altgeld Hall.

Friday, April 26, 2019

4:00 pm in 345 Altgeld Hall ,Friday, April 26, 2019

#### On generic monothetic subgroups of Polish groups

###### Dakota Ihli (UIUC Math)

Abstract: Given a Polish group $G$, what can be said about the subgroup $\overline{\left\langle g \right\rangle}$ for the generic element $g \in G$? In this talk we will discuss progress and open problems in this area. Special emphasis will be given on the group of measure-preserving automorphisms of the unit interval.

4:00 pm in 145 Altgeld Hall,Friday, April 26, 2019

#### Relatively hyperbolic groups and Dehn fillings

###### Heejoung Kim (UIUC)

Abstract: Geometric group theory has been studied extensively since Gromov introduced the notion of a hyperbolic group. For instance, the fundamental group of a hyperbolic surface is a hyperbolic group, but not the fundamental group of a cusped hyperbolic 3-manifold. From this motivating example, we consider a generalization of a hyperbolic group, called a relatively hyperbolic group. On the other hand, Thurston's Dehn filling theorem states that one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Groves and Manning extended Thurston's Dehn filling theorem to the context of relatively hyperbolic groups. In this talk, we will discuss hyperbolic groups, relatively hyperbolic groups, and the group-theoretic analog of Thurston's Dehn filling theorem in the context of relatively hyperbolic groups.

Monday, April 29, 2019

3:00 pm in 343 Altgeld Hall,Monday, April 29, 2019

#### Crystalline period map

###### Venkata Sai Bavisetty (UIUC Math)

Abstract: In Chromatic homotopy theory, one tries to understand the homotopy groups of spheres using the height filtration on formal group laws. This way at each height we get a spectral sequence whose $E_2$ term is the group cohomology of the Morava stabilizer group with coefficients in the Lubin-Tate ring. In this talk, I hope to explain a conceptual way to figure out the action of the Morava Stabilizer group on the Lubin-Tate ring.

3:00 pm in 243 Altgeld Hall,Monday, April 29, 2019

#### Symplectic capacities and the Minkowski sums of ellipsoids

###### Ely Kerman (UIUC)

Abstract: I will describe a new and elementary analysis of the Reeb flow on the boundary of the Minkowski sum of symplectic ellipsoids. This is made possible by an elegant parameterization of this boundary due to Chirikjian and Yan. Two immediate applications include an interesting manifestation of the "two to infinitely many" theorem of Hofer-Wysocki-Zehnder for closed Reeb orbits on strictly convex hyper surfaces, and a quick proof of the fact that the higher Ekeland-Hofer capacities, unlike the first one, fail to satisfy a Brun-Minkowski type inequality. This is a report on joint work in progress with Yuanpu Liang.

5:00 pm in 241 Altgeld Hall,Monday, April 29, 2019

#### Khalkali's notion of Ricci curvature in rotation algebras

###### Marius Junge (UIUC)

Abstract: As promised, we finish our exploration on Ricci curvature in operator algebras today. We may start 5minutes late.

Tuesday, April 30, 2019

12:00 pm in 243 Altgeld Hall,Tuesday, April 30, 2019

#### Topological Restrictions on Anosov Representations

###### Richard Canary (U Michigan)

Abstract: The theory of Anosov representations was introduced by Francois Labourie in his study of Hitchin representations. They have emerged as the natural analogue, for higher rank Lie groups, of Fuchsian representations, or more generally convex compact representations into rank one Lie groups. We will give a gentle introduction to Anosov representations, followed by a discussion of topological restrictions on the groups which admit Anosov representations into SL(d,R). For example, we will see characterizations of groups admitting Anosov representations into SL(3,R) and SL(4,R) and restrictions on the cohomological dimension for all values of d. (This is joint work with Kostas Tsouvalas.)

1:00 pm in 347 Altgeld Hall,Tuesday, April 30, 2019

#### Blow-up and Soliton Stability in KdV-type equations

###### Svetlana Roudenko (Florida International University)

Abstract: While the KdV equation and its generalizations with higher power nonlinearities (gKdV) have been long studied, a question about existence of blow-up solutions for higher power nonlinearities has posed lots of challenges and far from being answered. One of the main obstacles is that unlike other dispersive models such as the nonlinear Schrodinger or wave equations, the gKdV equation does not have a suitable virial quantity which is the key to prove the finite time blow-up. Partially, the question of existence and formation of singularities intertwines with the soliton stability or actually the instability, which may lead to a blowup. Only at the dawn of this century the groundbreaking works of Martel and Merle showed the existence of finite-time blow-up solutions for the quintic (critical) gKdV equation, as well as the asymptotic stability of solitons in the subcritical gKdV equation. We consider a higher dimensional extension of the gKdV equation, called generalized Zakharov-Kuznetsov (gZK) equation (the gKdV is limited as a spatially one-dimensional model), and investigate stability of solitons and the existence of blow-up solutions. We positively answer the question of existence of blowup in the two dimensional version of critical Zakharov-Kuznetsov equation and also obtain the asymptotic stability in the subcritical setting. We will discuss some of the important ingredients to obtain these results, including the Liouville-type theorem, which uses time-decay estimates, a la virial type quantity and spectral properties associated to it (this is a joint work with Luiz Farah, Justin Holmer and Kai Yang).

4:00 pm in 2 Illini Hall,Tuesday, April 30, 2019

#### Amenable first order theories

###### Anand Pillay (Notre Dame Math)

Abstract: (Joint with Hrushovski and Krupinski.) We extract from the properties (extreme) amenability of automorphisms groups of omega-categorical theories, the notion of (extreme) amenability of a first order theory T, which is much less restrictive than the automorphism property. I discuss some results around the Lascar group of such a theory, some proofs of which use versions of continuous logic.

Wednesday, May 1, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, May 1, 2019

#### A combinatorial proof of the pointwise ergodic theorem for actions of amenable groups along Tempelman Følner sequences

###### Jenna Zomback (UIUC)

Abstract: A pointwise ergodic theorem for the action of a countable group $\Gamma$ on a probability space equates the global ergodicity (atomicity) of the action to its pointwise combinatorics. Our main result is a short, combinatorial proof of the pointwise ergodic theorem for actions of amenable groups along Tempelman Følner sequences, which is a slightly less general version of Lindenstrauss's celebrated theorem. Without assuming any prior knowledge, we will work up to the general idea of the proof, which stems from Tserunyan's proof of the pointwise ergodic theorem for $\mathbb{Z}$ actions. This is joint work with Jon Boretsky.

Thursday, May 2, 2019

11:00 am in 241 Altgeld Hall,Thursday, May 2, 2019

#### The Distribution of log ζ(s) Near the Zeros of ζ

###### Fatma Cicek (Rochester Math)

2:00 pm in 147 Altgeld Hall,Thursday, May 2, 2019

#### Supnorm estimates for $\bar\partial$ in $\mathbb{C}^n$

###### Martino Fassina (Illinois Math)

Abstract: Let $\Omega$ be a domain in $\mathbb{C}^n$ and $f$ a $\bar\partial$-closed form on $\Omega$. A fundamental question in complex analysis is to establish the existence of solutions to the inhomogeneous Cauchy-Riemann equations $\bar\partial u=f$ that satisfy a norm estimate in $\Omega$. Whether such solutions exist depends both on the geometry of $\Omega$ and the regularity of $f$. In this talk, we consider the case where $\Omega$ is a polydisc. We establish the existence of weak solutions to $\bar\partial$ satisfying an $L^{\infty}$ estimate on $\Omega$ whenever the datum $f$ is in $L^{\infty}(\Omega)$, thus answering an old question of Kerzman and Stein. The talk is based on joint work with Yifei Pan.

3:00 pm in 347 Altgeld Hall,Thursday, May 2, 2019

#### Cell Decompositions for Rank Two Quiver Grassmannians

###### Dylan Rupel (Michigan State University)

Abstract: The (partial) flag varieties are among the most well-understood geometric objects. Essential to this understanding are the Schubert decompositions of these varieties. In this talk, I will recall two constructions of the Schubert decompositions of ordinary vector subspace Grassmannians and explain how analogues of these constructions combine to give rise to cell decompositions for Grassmannians of subrepresentations in (truncated) preprojective representations of acyclic quivers with two vertices. Depending on time I will describe some of the combinatorics underlying this geometry and discuss open problems and conjectures. This is a report on joint work with Thorsten Weist.

Friday, May 3, 2019

3:00 pm in 241 Altgeld Hall,Friday, May 3, 2019

###### Various (U of Illinois)

Abstract: In this panel, graduate students that have been involved in a variety of organizations will share their experiences as graduate student leaders. Through this panel, we will open the discussion on what are the ways we as graduate students can participate in leadership roles, what are some of the challenges we might face, and gain insights from established leaders in our community.

Saturday, May 4, 2019

4:00 pm in 243 Altgeld Hall,Saturday, May 4, 2019

#### To Be Announced

###### Partha Dey   [email] (University of Illinois at Urbana–Champaign)

Abstract: To Be Posted

Monday, May 6, 2019

3:00 pm in 243 Altgeld Hall,Monday, May 6, 2019

#### A two-category of Hamiltonian manifolds, and a (1+1+1) field theory

###### Guillem Cazassus (Indiana University)

Abstract: We define an extended field theory in dimensions 1+1+1, that takes the form of a 'quasi 2-functor' with values in a strict 2-category Hamˆ, defined as the 'completion of a partial 2-category' Ham, notions which we define. Our construction extends Wehrheim and Woodward's Floer Field theory, and is inspired by Manolescu and Woodward's construction of symplectic instanton homology. It can be seen, in dimensions 1+1, as a real analog of a construction by Moore and Tachikawa. Our construction is motivated by instanton gauge theory in dimensions 3 and 4: we expect to promote Hamˆ to a (sort of) 3-category via equivariant Lagrangian Floer homology, and extend our quasi 2-functor to dimension 4, via equivariant analogues of Donaldson polynomials.

Tuesday, May 7, 2019

1:00 pm in 345 Altgeld Hall ,Tuesday, May 7, 2019

#### An Intuitive Approach to the Martin Boundary

###### Peter Loeb (UIUC Math)

Abstract: The talk uses Robinson’s nonstandard analysis to give a rigorous, but intuitive, probabilistic construction of a compactifying boundary with maximal representing measures for positive harmonic functions.

3:00 pm in 243 Altgeld Hall,Tuesday, May 7, 2019

#### Construction of the Poincare sheaf on the stack of Higgs bundles

###### Mao Li (University of Wisconsin)

Abstract: An important part of the Langlands program is to construct the Hecke eignsheaf for irreducible local systems. Conjecturally, the classical limit of the Hecke eignsheaves should correspond to the Poincare sheaf on the stack of Higgs bundles. The Poincare sheaf for the compactified Jacobian of reduced planar curves have been constructed in the pioneering work of Dima Arinkin. In this talk I will present the construction of the Poincare sheaf on the stack of rank two Higgs bundles for any smooth projective curve over the entire Hitchin base, and it turns out to be a maximal Cohen-Macaulay sheaf. This includes the case of nonreduced spectral curves, and thus provides the first example of the existence of the Poincare sheaf for nonreduced planar curves.

4:00 pm in 243 Altgeld Hall,Tuesday, May 7, 2019

#### Some rigorous results on a long-range infection model on infinite lattices

###### Partha Dey   [email] (University of Illinois at Urbana–Champaign)

Abstract: We consider a long range infection model on finite dimensional lattices with no recovery and L\'evy interaction with exponent $\alpha$. Using monotonicity, one can prove that the growth rate decreases as $\alpha$ increases. We prove existence of four different growth regimes with thresholds depending on the dimension:
a) for $\alpha$ < d instantaneous growth,
b) for d < $\alpha$ < 2d stretched exponential growth,
c) for 2d< $\alpha$ < 2d+1 super linear growth and
d) for $\alpha$>2d+1 linear growth,
where $d$ is the dimension. In one-dimension we characterize the asymptotic distributional limits, which shows existence of a new fluctuation'' transition threshold. Finally, we will mention partial results and conjectures in higher dimension and when recovery is added to the model. Prior knowledge about epidemics models is not required for this talk.

Monday, May 13, 2019

3:00 pm in 243 Altgeld Hall,Monday, May 13, 2019

#### Kirwan-Ness stratifications in algebraic geometry

###### Itziar Ochoa (Yale University)

Abstract: Given an algebraic variety $X$ with an action of a reductive group $G$, geometric invariant theory splits $X$ as the disjoint union $X=X^{ss}\sqcup X^{un}$ of the semistable and unstable locus. The Kirwan-Ness stratification refines $X$ even more by describing $X^{un}$ as a disjoint union of strata $X^{un}=\displaystyle\sqcup_{\beta\in\textsf{KN}} S_\beta$ determined by 1-parameter subgroups $\beta$. In this talk we will describe an algorithm that finds the $\beta$'s and show that such algorithm can be simplified when our space is of the form $T^*V$ where $V$ is a vector space.

Thursday, May 16, 2019

#### Actuarial Science Reunion

Abstract: The Risk Analytical Symposium will be held from 8:30 am - 4:30 pm at the Illini Center, 200 S. Wacker Drive, Chicago. The Illinois Actuarial Science Reception will be held from 4-7 pm at the Deloitte Office Building, 29th Floor, 111 S. Wacker Drive, Chicago. To register go to math.illinois.edu/illinois-actuarial-science

Sunday, May 19, 2019

2:00 pm in South Lounge, Illini Union,Sunday, May 19, 2019

#### Department of Mathematics Retirement Reception

Abstract: The Department of Mathematics will hold a retirement reception from 2-4 pm on Sunday, May 19, 2019, in the South Lounge of the Illini Union. Please join us as we honor the following individuals: Maarten Bergvelt, Bruce Berndt, Tori Corkery, Julian Palmore, Zhong-Jin Ruan, and Jang-Mei Wu.

Tuesday, May 21, 2019

11:00 am in 241 Altgeld Hall,Tuesday, May 21, 2019

#### Involution graph Schubert varieties

###### Brendan Pawlowski (University of Southern California)

Abstract: Every product of Schur functions arises as a Stanley symmetric function, which can be decomposed into Schur terms by Lascoux and Schutzenberger's transition recurrence, giving a Littlewood-Richardson rule as a special case. A similar approach using involution Stanley symmetric functions gives a Littlewood-Richardson rule for Schur P- and Q-functions. On the geometric side, Knutson, Lam, and Speyer showed that Stanley symmetric functions represent cohomology classes of graph Schubert varieties in Grassmannians. We identify analogous varieties in isotropic Grassmannians whose classes correspond to involution Stanley symmetric functions. We hope that this geometric connection will allow the transition approach to Littlewood-Richardson rules to be extended to the K-theory (and other cohomology theories) of isotropic Grassmannians.

Tuesday, May 28, 2019

2:00 pm in 243 Altgeld Hall,Tuesday, May 28, 2019

#### Stallings' foldings and cost of treeable equivalence relations (Part 1)

###### Anush Tserunyan (UIUC)

Abstract: Introduced by Levitt and extensively developed by Gaboriau, cost is a very useful real-valued invariant for probability measure preserving countable equivalence relations that measures the infimum amount of edges needed to connect a.e. equivalence class in a uniformly Borel fashion. We discuss Gaboriau's original proof of his Fundamental Theorem of Cost, which states that the cost of a treeable equivalence relation is achieved by any Borel treeing of it.

Wednesday, May 29, 2019

2:00 pm in 243 Altgeld Hall,Wednesday, May 29, 2019

#### Stallings' foldings and cost of treeable equivalence relations (Part 2)

###### Anush Tserunyan (UIUC)

Abstract: Introduced by Levitt and extensively developed by Gaboriau, cost is a very useful real-valued invariant for probability measure preserving countable equivalence relations that measures the infimum amount of edges needed to connect a.e. equivalence class in a uniformly Borel fashion. We discuss Gaboriau's original proof of his Fundamental Theorem of Cost, which states that the cost of a treeable equivalence relation is achieved by any Borel treeing of it.

Thursday, June 6, 2019

4:00 pm in 100 Noyes Lab ,Thursday, June 6, 2019

#### Ramanujan--The Ultimate Superhero

###### Bruce Berndt (Illinois Math)

Abstract: Srinivasa Ramanujan is perhaps the most enigmatic mathematician in the history of our subject. First, a short account of Ramanujan’s life will be given. Second, the speaker will provide a history of his association with Ramanujan beginning with Robert Rankin in 1966, Paul Bateman in 1967, Emil Grosswald in 1970, and a cold office at the Institute for Advanced Study in February, 1974. Third, some examples from Ramanujan’s Notebooks and Lost Notebook will be given to provide evidence that this great Indian mathematician is, indeed, The Ultimate Superhero. This lecture will be aimed at a general audience.

Thursday, June 13, 2019

3:00 pm in 243 Altgeld Hall,Thursday, June 13, 2019

#### Wilf's conjecture by multiplicity

###### Winfried Bruns (Mathematik/Informatik Universität Osnabrück)

Abstract: Let S be a numerical semigroup. Its embedding dimension e(S) is the minimal number of generators, the Frobenius number F(S) is the largest integer not in S , and n(S) counts the elements in S that are < F(S). Wilf's conjecture states that F(S) < e(S)n(S). It has been proved in many cases, but remains a major open problem in the combinatorial theory of numerical semigroups. We will show that for fixed multiplicity m=m(S), the smallest nonzero element of S, the conjecture can be decided algorithmically by polyhedral methods using the parametrization of multiplicity m semigroups by the lattice points of the Kunz polyhedron P(m). With them we have verified the conjecture for m up to 18.

Friday, June 14, 2019

2:00 pm in 243 Altgeld Hall,Friday, June 14, 2019

#### Decomposing graphs into edges and triangles

###### Bernard Lidicky (Iowa State Math)

Abstract: We prove the following 30-year old conjecture of Gyori and Tuza: the edges of every $n$-vertex graph $G$ can be decomposed into complete graphs $C_1,\ldots,C_\ell$ of orders two and three such that $|C_1|+\cdots+|C_\ell|\le (1/2+o(1))n^2$. This result implies the asymptotic version of the old result of Erdos, Goodman and Posa that asserts the existence of such a decomposition with $\ell\le n^2/4$. We also discuss removing $o(1)$ term sharpening the result and possible extensions. The talk is based on joint works with Blumenthal, Kral, Martins, Pehova, Pikhurko, Pfender, Vole.

Friday, August 9, 2019

3:00 pm in 345 Altgeld Hall,Friday, August 9, 2019

#### Atomic decomposition of characters and crystals

###### Cristian Lenart (University at Albany, SUNY)

Abstract: A. Lascoux stated that the type $A$ Kostka-Foulkes polynomials $K_{\lambda,\mu}(t)$ expand positively in terms of so-called atomic polynomials. For any semisimple Lie algebra, the former polynomial is a $t$-analogue of the multiplicity of the dominant weight $\mu$ in the irreducible representation of highest weight $\lambda$. I formulate the atomic decomposition in arbitrary type, and also define a combinatorial version of it, as a decomposition of a modified version of the Kashiwara crystal graph encoding the representation. This stronger version is shown to hold in type $A$ (which provides a new, conceptual approach to Lascoux's statement), as well as in types $B$, $C$, and $D$ in a stable range for $t=1$. Some applications are also discussed. This is joint work with Cedric Lecouvey, and the presentation will be largely self-contained.

Tuesday, August 20, 2019

2:00 pm in 343 Altgeld Hall,Tuesday, August 20, 2019

#### Refinements of Choice Number of Graphs

###### Xuding Zhu (Zheiang Normal University Math)

Abstract: In this talk, we discuss some refinements of choice number of graphs.
(1) We define a graph $G$ to be strongly fractional $r$-choosable if for any positive integer $m$, $G$ is $(\lceil rm \rceil, m)$-choosable, and define the strong fractional choice number $ch^∗_f(G)$ of $G$ to be the infimum $r$ for which G is strongly fractional $r$-choosable. The strong fractional choice number of a family $\mathcal G$ of graphs is defined to be the supremum of $ch^∗_f(G)$ for graph $G \in \mathcal G$. It is proved that the strong fractional choice number of planar graphs is at least $4+2/9$, and the strong fractional choice number of triangle free planar graphs is at least $3 + 1/17$.
(2) We say a graph $G$ is $(k + \epsilon)$-choosable if any subgraph $H$ of $G$ has a subset $X$ of vertices for which the following hold: (i) $|X| \le \epsilon|V (H)|$, (ii) for any list assignment $L$ of $G$ for which $|L(v)| = k+1$ for $v \in X$ and $|L(v)| = k$ for $v \in V (H)−X$, $H$ is $L$-colourable. It is proved that planar graphs are $(4 + 1/2)$-choosable and triangle free planar graphs are $(3 + 2/3)$-choosable.
(3) Assume $\lambda = (k_1, k_2, \dots, k_q)$ is a partition of an integer $k$. A $\lambda$-assignment of $G$ is a $k$-assignment $L$ of $G$ for which the colour set $\bigcup_{v \in V(G)} L(v)$ can be partitioned into $C_1 \cup C_2 \cup \dots \cup C_q$ such that $|L(v) \cap C_i| = k_i$ for every vertex $v$. We say $G$ is $\lambda$-choosable if $G$ is $L$-colourable for every $\lambda$-assignment $L$ of $G$. If $\lambda = \{k\}$, then $\lambda$-choosability is the same as $k$-choosability; if $\lambda = \{1,1,\dots,1\}$ then $\lambda$-choosability is the same as $k$-colourability. For other partitions $\lambda$ of $k$, $\lambda$-choosability form a complex hierachy of complexity of colourability. A recent result of Kermnitz and Voigt implies that for any partition $\lambda$ of $4$ other than $\{1,1,1,1\}$, there is a planar graph which is not $\lambda$-choosable (this is much stronger than the result that there are planar graphs that are not $4$-choosable). Some basic properties of $\lambda$-choosability and relation to some other colouring concepts will be discussed.

Monday, August 26, 2019

3:00 pm in 441 Altgeld Hall,Monday, August 26, 2019

#### Organizational Meeting

###### Brian Shin (UIUC Math)

Tuesday, August 27, 2019

12:00 pm in 243 Altgeld Hall,Tuesday, August 27, 2019

#### Views of the Scenery Flow

###### Albert Fisher (University of São Paulo)

Abstract: I will give an overview of work carried out together with T. Bedford, M. Urbanski, P. Arnoux, and M. Talet. We consider the result of zooming toward a point of a geometric object embedded in an ambient Euclidean space. For a smooth submanifold this just converges to the tangent space at the point, a fixed point for the flow. But for irregular objects like fractal sets the scenery at small scales keeps changing. The collection of the asymptotic limiting objects can be thought of as the tangent bundle of the fractal, these scenes'' are related to Furstenberg's microsets''. This collection of sets is acted upon by the flow of zooming exponentially fast toward small scales. A mathematical challenge, then, is to precisely define this scenery flow'' and to study its dynamical properties. As we sketch, for the limit set of a Kleinian group, the scenery flow is isomorphic to the geodesic frame flow of the associated hyperbolic three-manifold. Now as Sullivan proved, this flow entropy equals the Hausdorff dimension of the limit set. On the other hand, Bowen showed the Hausdorff dimension of the limit set is given in terms of a zero of the pressure function for log of the derivative. This formula for dimension was extended by Ruelle to hyperbolic Julia sets. With Bedford and Urbanski, we constructed the scenery flow for hyperbolic Julia sets, leading to a unification of Bowen's formula with Sullivan's: in both cases, "Hausdorff dimension equals scenery flow entropy". In this sense the scenery flow thus provides an analogue for Julia sets of the geodesic flow. These ideas extend to other situations. In work with M. Talet and P. Arnoux we have studied the scenery flow for Brownian motion paths, and for the nested tilings associated with circle rotations and interval exchange transformations and defined by renormalization. We sketch these ideas, indicating the methods of proof.

1:00 pm in 345 Altgeld Hall,Tuesday, August 27, 2019

#### Organizational meeting

1:00 pm in Altgeld Hall,Tuesday, August 27, 2019

#### Deflated Continuation: A bifurcation analysis tool for Nonlinear Schrodinger (NLS) Systems

###### Stathis Charalampidis (Mathematics Department, California Polytechnic State University)

Abstract: Continuation methods are numerical algorithmic procedures for tracing out branches of fixed points/roots to nonlinear equations as one (or more) of the free parameters of the underlying system is varied. On top of standard continuation techniques such as the sequential and pseudo-arclength continuation, we will present a new and powerful continuation technique called the deflated continuation method which tries to find/construct undiscovered/disconnected branches of solutions by eliminating known branches. In this talk we will employ this method and apply it to the one-component Nonlinear Schrodinger (NLS) equation in two spatial dimensions. We will present novel nonlinear steady states that have not been reported before and discuss bifurcations involving such states. Next, we will focus on a two-component NLS system and discuss about recent developments by using the deflated continuation method where the landscape of solutions of such a system is far richer. A discussion about the challenges in the two-component setting will be offered and a summary of open problems will be emphasized.

2:00 pm in 243 Altgeld Hall,Tuesday, August 27, 2019

#### The Alon-Tarsi number of subgraphs of a planar graph

###### Seog-Jin Kim (Konkuk University Math)

Abstract: This paper constructs a planar graph $G_1$ such that for any subgraph $H$ of $G_1$ with maximum degree $\Delta(H) \le 3$, $G_1-E(H)$ is not $3$-choosable, and a planar graph $G_2$ such that for any star forest $F$ in $G_2$, $G_2-E(F)$ contains a copy of $K_4$ and hence $G_2-E(F)$ is not $3$-colourable. On the other hand, we prove that every planar graph $G$ contains a forest $F$ such that the Alon-Tarsi number of $G - E(F)$ is at most $3$, and hence $G - E(F)$ is 3-paintable and 3-choosable. This is joint work with Ringi Kim and Xuding Zhu.

3:00 pm in 243 Altgeld Hall,Tuesday, August 27, 2019

#### Organizational Meeting

Wednesday, August 28, 2019

3:30 pm in 241 Altgeld Hall,Wednesday, August 28, 2019

#### Organizational meeting

4:00 pm in Altgeld Hall 447,Wednesday, August 28, 2019

#### Organizational Meeting

Thursday, August 29, 2019

2:00 pm in 347 Altgeld Hall,Thursday, August 29, 2019

#### Organizational Meeting

Friday, August 30, 2019

1:00 pm in 141 Altgeld Hall,Friday, August 30, 2019

#### Organizational Meeting

###### Kesav Krishnan (Illinois Math)

Abstract: This meeting will be to decide an optimal time for weekly meetings, as well as discuss a tentative schedule for speakers

2:00 pm in 347 Altgeld Hall,Friday, August 30, 2019

#### Organizational Meeting

###### Chelsea Walton (UIUC)

Abstract: This will be a short organizational meeting to introduce the structure of the Algebra seminar. Here is the seminar website: https://faculty.math.illinois.edu/~notlaw/UIUC-Algebra.html. Half of the talk slots will be for (introductory +) research talks, and the other half of the slots will be dedicated to a reading group on Tensor Categories. The format of the reading group will be similar to that for the TQFT reading group in Spring 2019 (https://faculty.math.illinois.edu/~notlaw/teaching.html#past), except that the duration of talk slots will be longer.

3:00 pm in 343 Altgeld Hall,Friday, August 30, 2019

#### Organizational Meeting

Abstract: This will be the organizational meeting for the graduate student number theory seminar. We will discuss the schedule for weekly meetings, as well as begin sign-up for speakers.

4:00 pm in 141 Altgeld Hall,Friday, August 30, 2019

#### Organizational Meeting

Tuesday, September 3, 2019

12:50 pm in 347 Altgeld Hall,Tuesday, September 3, 2019

#### Multi-frequency class averaging for three-dimensional cryo-electron microscopy

###### Zhizhen Jane Zhao (Illinois ECE)

Abstract: We introduce a novel intrinsic classification algorithm--multi-frequency class averaging (MFCA)--for clustering noisy projection images obtained from three-dimensional cryo-electron microscopy (cryo-EM) by the similarity among their viewing directions. This new algorithm leverages multiple irreducible representations of the unitary group to introduce additional redundancy into the representation of the transport data, extending and outperforming the previous class averaging algorithm that uses only a single representation. We will discuss the formal algebraic model and representation theoretic patterns of the proposed MFCA algorithm. We conceptually establish the consistency and stability of MFCA by inspecting the spectral properties of a generalized localized parallel transport operator on the two-dimensional unit sphere through the lens of Wigner D-matrices. We will also show how this algorithm can be applied to directly denoise the real data.

1:00 pm in 345 Altgeld Hall,Tuesday, September 3, 2019

#### Ergodic theorems and more fun with countable Borel equivalence relations

###### Jenna Zomback (UIUC Math)

Abstract: I will discuss my current and future work, which lies in the study of countable Borel equivalence relations (CBERs) and its applications to ergodic theory and measured group theory. In the first section of the talk, I will discuss a tiling result for amenable groups along Tempelman Følner sequences and explain how this result implies the corresponding pointwise ergodic theorem (this is joint work with Jonathan Boretsky). In the second section, I will introduce the notion of cost of an equivalence relation, and state a few important results in this field. In each of the two sections, I will state some proposed avenues for future work. This talk is part of a preliminary examination.

2:00 pm in 243 Altgeld Hall,Tuesday, September 3, 2019

#### Large Monochromatic Components in Sparse Random Hypergraphs

###### Sean English (Illinois Math)

Abstract: It is known, due to Gyárfás and Füredi, that for any $r$-coloring of the edges of $K_n$, there is a monochromatic component of order $(1/(r-1)+o(1))n$. Recently, Bal and DeBiasio, and independently Dudek and Prałat showed that the Erdős-Rényi random graph $\mathcal{G}(n,p)$ behaves very similarly with respect to the size of the largest monochromatic component. More precisely, it was shown that a.a.s. for any $r$-coloring of the edges of $\mathcal{G}(n,p)$ and arbitrarily small constant $\alpha>0$, there is a monochromatic component of order $(1/(r-1)-\alpha)n$, provided that the average degree goes to infinity with $n$. As before, this result is best possible.

In this talk we present a generalization of this result to hypergraphs. Specifically we show that in the $k$-uniform random hypergraph, $\mathcal{H}^{(k)}(n,p)$ a.a.s. for any $k$-coloring of the edges, there is a monochromatic component of order $(1-\alpha)n$. Furthermore, for any $k+1$ coloring, there is a monochromatic component of order $(1-\alpha)\frac{k}{k+1}n$. These results hold as long as the average degree goes to infinity.

It is also known Gyárfás, Sárközy and Szemerédi that the Ramsey number for loose cycles on $n$ vertices in $k$-uniform hypergraphs is asymptotically $\frac{2k-1}{2k-2}n$, which implies that in any $2$-coloring of $K^{(k)}_n$, for large $n$, we can find a loose cycle on about $\frac{2k-2}{2k-1}n$ vertices. We will present a generalization of this which shows that even if the host graph is $\mathcal{H}^{(k)}_{n,p}$, this result still holds a.a.s. provided that the average degree goes to infinity.

This project is joint work with Patrick Bennett, Louis Debiasio and Andrzej Dudek.

Wednesday, September 4, 2019

4:00 pm in Altgeld Hall 447,Wednesday, September 4, 2019

#### Counting rational curves on K3 surfaces and modular forms

###### Sungwoo Nam (Illinois Math)

Abstract: Curve counting invariants on K3 surfaces turn out to have an interesting connection to modular forms via Yau-Zaslow formula. In this talk, starting from the basic properties of K3 surfaces, I’ll discuss two proofs of Yau-Zaslow formula due to Beauville which uses Euler characteristic of compactified Jacobian, and Bryan-Leung using Gromov-Witten technique. If time permits, I’ll describe generalizations of the formula such as Göttsche’s formula and Katz-Klemm-Vafa formula.

Thursday, September 5, 2019

11:00 am in 241 Altgeld Hall,Thursday, September 5, 2019

#### On the modularity of elliptic curves over imaginary quadratic fields

###### Patrick Allen (Illinois)

Abstract: Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point, implying that the mod 3 Galois representation attached to the elliptic curve arises from a modular form of weight one. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch" that gives a criterion for when a given mod 6 representation arises from an elliptic curve. As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.

12:00 pm in 243 Altgeld Hall ,Thursday, September 5, 2019

#### Counting incompressible surfaces in 3-manifolds

###### Nathan Dunfield   [email] (Illinois)

Abstract: Counting embedded curves on a hyperbolic surface as a function of their length has been much studied by Mirzakhani and others. I will discuss analogous questions about counting incompressible surfaces in a hyperbolic 3-manifold, with the key difference that now the surfaces themselves have intrinsic topology. As are only finitely many incompressible surfaces of bounded Euler characteristic up to isotopy in a hyperbolic 3-manifold, it makes sense to ask how the number of isotopy classes grows as a function of the Euler characteristic. Using Haken’s normal surface theory and facts about branched surfaces, we can characterize not just the rate of growth but show it is (essentially) a quasi-polynomial. Moreover, our method allows for explicit computations in reasonably complicated examples. This is joint work with Stavros Garoufalidis and Hyam Rubinstein.

1:00 pm in 464 Loomis,Thursday, September 5, 2019

#### FROM LOCALITY TO NON-LOCALITY: FERMIONIC ENTANGLEMENT ON THE TORUS

###### Ignacio A. Reyes (Max Planck Institute, Potsdam)

Abstract: We uncover various novel aspects of the entanglement of free fermions at finite temperature on the circle. The modular flow involves a bi-local coupling between a discrete but infinite set of points, even for a single interval. The modular Hamiltonian transitions from locality to complete non-locality as a function of temperature. We derive the entanglement and relative entropies, and comment on the applications to bulk reconstruction in higher spin holography.

2:00 pm in 347 Altgeld Hall,Thursday, September 5, 2019

#### Introduction to Random Planar Maps

###### Grigory Terlov (UIUC Math)

Abstract: A "typical" continuous curve on a plane looks like a path of Brownian motion. A natural next question we might ask is "what does a "typical" continuous 2d-surface looks like?" One of the ways to construct such a model is to find a discrete object and consider a scaling limit of it (analogous to considering a scaling limit of a random walk to construct Brownian motion). Such objects are called random planar maps - planar multi-graphs embedded in a sphere or a plane. Of course, similarly to random walks, there are many other reasons why these discrete objects are interesting. In these two talks we will consider several ways of defining random planar maps and a measure on them, connections with random walks and random trees. Finally, in the remaining time I will try to mention several highlights of the field in connection with combinatorics, percolation theory, scaling limits, and Ergodic theory.

3:00 pm in 347 Altgeld Hall,Thursday, September 5, 2019

#### Polytopes, polynomials and recent results in 1989 mathematics

###### Bruce Reznick   [email] (University of Illinois at Urbana-Champaign)

Abstract: Hilbert’s 17th Problem discusses the possibility of writing polynomials in several variables which only take non-negative values as a sum of squares of polynomials. One approach is to substitute squared monomials into the arithmetic-geometric inequality. Sometimes this is a sum of squares, sometimes it isn’t, and I proved 30 years ago that this depends on a property of the polytope whose vertices are the exponents of the monomials in the substitution. What’s new here is an additional then-unproved claim in that paper and its elementary, but non-obvious proof. This talk lies somewhere in the intersection of combinatorics, computational algebraic geometry and number theory and is designed to be accessible to first year graduate students.

Friday, September 6, 2019

2:00 pm in 347 Altgeld Hall,Friday, September 6, 2019

#### Quantum Symmetry in the context of co/representation categories

###### Chelsea Walton (UIUC)

Abstract: In the first 50 minutes, I will provide an introductory talk on quantum symmetry in the context of co/representation categories, which serves as one point of motivation for the reading group on Tensor Categories. I will then give a follow-up research talk in the second 50 minutes on joint work in progress with Elizabeth Wicks and Robert Won on quantum symmetry and weak Hopf algebras.

3:00 pm in 341 Altgeld Hall,Friday, September 6, 2019

#### Introduction to Metric Embeddings into Banach Spaces

###### Chris Gartland (UIUC)

Abstract: This talk will survey some results on the existence or nonexistence of embeddings of certain metric spaces into Banach spaces. Some proofs will be provided whenever sufficiently elementary, but most results will only be stated. There is a wealth of tools used in this field, and we'll encounter results whose proofs could include graph-theoretical combinatorics or abstract harmonic analysis. Topics to be covered (time permitting of course) range from: elementary facts - every finite metric space isometrically embeds into $\ell^\infty$, to graduate level analysis - almost everywhere differentiation of absolutely continuous functions, and finally to a recent, deep application in computer science - a full solution to the Goemans-Linial conjecture. Families of expander graphs and the Heisenberg group are metric spaces that play a special role in this story.

3:00 pm in Illini Hall 1,Friday, September 6, 2019

#### Series and Polytopes

###### Vivek Kaushik (Illinois Math)

Abstract: Consider the series $S(k)=\sum_{n \geq 0} \frac{(-1)^{nk}}{(2n+1)^k}$ for $k \in \mathbb{N}.$ It is well-known that $S(k)$ is a rational multiple of $\pi^k$ using standard techniques from either Fourier Analysis or Complex Variables. But in this talk, we evaluate $S(k)$ through multiple integration. On one hand, we start with a $k$-dimensional integral that is equal to the series in question. On the other hand, a trigonometric change of variables shows the series is equal to the volume of a convex polytope in $\mathbb{R}^k.$ This volume is proportional to a probability involving certain pairwise sums of $k$ independent uniform random variables on $(0,1).$ We obtain this probability using combinatorial analysis and multiple integration, which ultimately leads to us finding an alternative, novel closed formula of $S(k).$

4:00 pm in 345 Altgeld Hall,Friday, September 6, 2019

#### "On the nonexistence of Følner sets" by Isaac Goldbring

###### Elliot Kaplan (UIUC Math)

Abstract: This will be the first (and possibly only) talk on the preprint "On the nonexistence of Følner sets" by Isaac Goldbring (https://arxiv.org/abs/1901.02445). I will introduce all of the necessary model-theoretic and group-theoretic background. Time permitting, I may get to the proof of the main result.

4:00 pm in 141 Altgeld Hall,Friday, September 6, 2019

#### Geometry by example: the projective plane

Abstract: In this expository talk, I will introduce the projective plane, and use it to explore a range of ideas including moment polytopes, localization formulas and intersection theory.

Monday, September 9, 2019

3:00 pm in 243 Altgeld Hall,Monday, September 9, 2019

#### Organizational Meeting

3:00 pm in 441 Altgeld Hall,Monday, September 9, 2019

#### Algebraic theories and homotopy theory

###### William Balderrama (UIUC Math)

Abstract: In this talk, I will motivate and introduce algebraic theories as a category-theoretic approach to finite product theories. I will then talk about a well-behaved notion of an infinitary algebraic theory, and the introduction of homotopy-theoretic structure, which can be used to define notions of homology and cohomology for the models of an algebraic theory. This is the first of two talks; the second will use these ideas to produce applications in stable homotopy theory.

5:00 pm in 241 Altgeld Hall,Monday, September 9, 2019

#### What can be in an IGL project on C*-algebras?

###### Chris Linden (UIUC)

Abstract: Report on IGL project in C*-algebras by Chris Linden

Tuesday, September 10, 2019

11:00 am in 347 Altgeld Hall,Tuesday, September 10, 2019

#### Dirichlet character twisted Eisenstein series and $J$-spectra

###### Ningchuan Zhang

Abstract: Bernoulli numbers show up in both the $q$-expansions of normalized Eisenstein series and the image of the $J$-homomorphism in the stable homotopy groups of spheres. Number theorists have defined generalized Bernoulli numbers and twisted Eisenstein series associated to Dirichlet characters. The goal of this talk is to construct a family of Dirichlet character twisted $J$-spectra and explain the relations between their homotopy groups and congruences of the twisted Eisenstein series. In the course of that, we will generalize Nicholas Katz’s algebro-geometric explanation of congruences of the (untwisted) normalized Eisenstein series in his Antwerp notes.

1:00 pm in 345 Altgeld Hall,Tuesday, September 10, 2019

###### Anush Tserunyan (UIUC Math)

Abstract: Dating back to Birkhoff, pointwise ergodic theorems for probability measure preserving (pmp) actions of countable groups are bridges between the global condition of ergodicity (measure-theoretic transitivity) and the local combinatorics of the actions. Each such action induces a Borel equivalence relation with countable classes and the study of these equivalence relations is a flourishing subject in modern descriptive set theory. Such an equivalence relation can also be viewed as the connectedness relation of a locally countable Borel graph. These strong connections between equivalence relations, group actions, and graphs create an extremely fruitful interplay between descriptive set theory, ergodic theory, measured group theory, probability theory, and descriptive graph combinatorics. I will discuss how descriptive set theoretic thinking combined with combinatorial and measure-theoretic arguments yields a pointwise ergodic theorem for quasi-pmp locally countable graphs. This theorem is a general random version of pointwise ergodic theorems for group actions and is provably the best possible pointwise ergodic result for some of these actions.

2:00 pm in 243 Altgeld Hall,Tuesday, September 10, 2019

#### The largest projective cube-free subsets of $\mathbb Z_{2^n}$

###### Adam Zsolt Wagner (ETH Zurich Math)

Abstract: What is the largest subset of $\mathbb Z_{2^n}$ that doesn't contain a projective $d$-cube? In the Boolean lattice, Sperner's, Erdos's, Kleitman's and Samotij's theorems state that families that do not contain many chains must have a very specific layered structure. We show that if instead of $\mathbb Z_2^n$ we work in $\mathbb Z_{2^n}$, analogous statements hold if one replaces the word $k$-chain by projective cube of dimension $2^{k-1}$. The largest $d$-cube-free subset of $\mathbb Z_{2^n}$, if $d$ is not a power of two, exhibits a much more interesting behaviour.

(Joint work with Jason Long)

3:00 pm in 243 Altgeld Hall,Tuesday, September 10, 2019

#### Deformation theory and partition Lie algebras

###### Akhil Mathew (U Chicago)

Abstract: A theorem of Lurie and Pridham states that over a field of characteristic zero, derived "formal moduli problems" (i.e., deformation functors defined on derived Artinian commutative rings), correspond precisely to differential graded Lie algebras. This formalizes a well-known philosophy in deformation theory, and arises from Koszul duality between Lie algebras and commutative algebras. I will report on joint work with Lukas Brantner, which studies the analogous situation for arbitrary fields. The main result is that formal moduli problems are equivalent to a category of "partition Lie algebras"; these are algebraic structures (which agree with DG Lie algebras in characteristic zero) which arise from a monad built from the partition complex.

Wednesday, September 11, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, September 11, 2019

#### On spatial actions and whirly actions

###### Dakota Ihli (UIUC Math)

Abstract: Given a Borel measure-preserving action of a Polish group $G$ on $\left[ 0,1 \right]$, ignoring null sets gives a corresponding action on the measure algebra. If $G$ is locally compact, this can always be reversed: a measure-preserving action on the measure algebra of $\left[ 0,1 \right]$ can always be induced by a (pointwise) action on $\left[ 0,1 \right]$. This fails in the non-locally-compact case. In this talk, I will discuss a dynamical condition on the action ("whirliness") which characterizes when an action on the measure algebra admits a pointwise realization. This talk is based on 'Spatial and non-spatial actions of Polish groups' by E Glasner and B Weiss.

4:00 pm in 245 Altgeld Hall,Wednesday, September 11, 2019

#### Trees and leaves where boundaries meet

###### Elizabeth Field   [email] (Illinois Math)

Abstract: If $H$ and $G$ are hyperbolic groups with $H\leq G$, one can ask if the inclusion map from $H$ into $G$ extends continuously to a map from the boundary of $H$ into the boundary of $G$. If such a map exists, we call this map the Cannon-Thurston map. In this talk, we will first draw inspiration from the setting originally studied by Cannon and Thurston which has a particularly nice geometry. We will then take a brief tour through the world of geometric group theory, where we will discuss the notion of a hyperbolic group and its boundary. Finally, we will return to explore various geometric, topological, and algebraic properties of the Cannon-Thurston map.

4:00 pm in 447 Altgeld Hall,Wednesday, September 11, 2019

#### Compactified Jacobian

###### Lutian Zhao (Illinois Math)

Abstract: In this talk, I will start by a brief review of the history of Jacobians. Then I will describe the definition of compactified Jacobian as they are crucial object in studying singular curves. The final goal is to understand a calculation on the compactified Jacobian of the curve $x^p-y^q=0$ for $p,q$ coprime, where the Euler characteristic of the Compactified Jacobian is exactly the Catalan number $C_{p,q}$.

Thursday, September 12, 2019

11:00 am in 241 Altgeld Hall,Thursday, September 12, 2019

#### Moments of half integral weight modular L–functions, bilinear forms and applications

###### Alexander Dunn (Illinois Math)

Abstract: Given a half-integral weight holomorphic newform $f$, we prove an asymptotic formula for the second moment of the twisted L-function over all primitive characters modulo a prime. In particular, we obtain a power saving error term and our result is unconditional; it does not rely on the Ramanujan-Petersson conjecture for the form $f$. This gives a very sharp Lindelöf on average result for L-series attached to Hecke eigenforms without an Euler product. The Lindelöf hypothesis for such series was originally conjectured by Hoffstein. In the course of the proof, one must treat a bilinear form in Salié sums. It turns out that such a bilinear form also has several arithmetic applications to equidistribution. These are a series of joint works with Zaharescu and Shparlinski-Zaharescu.

12:00 pm in 243 Altgeld Hall,Thursday, September 12, 2019

#### Weil-Petersson translation length and manifolds with many fibered fillings.

###### Chris Leininger (Illinois Math)

Abstract: In this talk, I will discuss joint work with Minsky, Souto, and Tayor in which we prove that any mapping torus of a pseudo-Anosov mapping class with bounded normalized Weil-Petersson (WP) translation length contains a finite set of “vertical and horizontal closed curves”, and drilling out this set of curves results in one of a finite number of cusped hyperbolic 3–manifolds (depending only on the normalized WP length bound). This echoes an earlier result, joint with Farb and Margalit, for the Teichmuller metric. We also prove new estimates for the WP translation length of compositions of pseudo-Anosov mapping classes and arbitrary powers of a Dehn twist.

1:00 pm in 464 Loomis,Thursday, September 12, 2019

#### EXPERIMENTS WITH MACHINE LEARNING IN STRING GEOMETRY

###### Vishnu Jejjala (University of Witwatersrand)

Abstract: Identifying patterns in data enables us to formulate questions that can lead to exact results. Since many of the patterns are subtle, machine learning has emerged as a useful tool in discovering these relationships. We show that topological features of Calabi–Yau geometries are machine learnable. We indicate the broad applicability of our methods to existing large data sets by finding relations between knot invariants, in particular, the hyperbolic volume of the knot complement and the Jones polynomial.

2:00 pm in 347 Altgeld Hall,Thursday, September 12, 2019

#### Introduction to Random Planar Maps part 2

###### Grigory Terlov (UIUC Math)

Abstract: The main focus of the second part of this talk is to discuss bisections between random bipartite planar maps and decorated Galton Watson trees. Then if time permits we will continue connecting this model with other areas of probability that audience might be familiar with and/or interested in exploring.

4:00 pm in 245 Altgeld Hall,Thursday, September 12, 2019

#### Non-Smooth Harmonic Analysis

###### Palle Jorgensen   [email] (University of Iowa)

Abstract: While the framework of the talk covers a wider view of harmonic analysis on fractals, it begins with a construction by the author of explicit orthogonal Fourier expansions for certain fractals. It has since branched off several directions, each one dealing with aspect of the wider subject. The results presented cover (among other papers) joint work with Steen Pedersen, then later, with Dorin Dutkay. Fractals. Intuitively, it is surprising that any selfsimilar fractals in fact do admit orthogonal Fourier series. And our initial result generated surprised among members of the harmonic analysts community. The theme of Fourier series on Fractals has by now taken off in a number of diverse directions; e.g., (i) wavelets on fractals, or frames; (ii) non-commutative analysis on graph limits, to mention only two. Two popular question are: “What kind of fractals admit Fourier series?” “If they don’t, then what alternative harmonic analysis might be feasible?”

Friday, September 13, 2019

2:00 pm in 347 Altgeld Hall,Friday, September 13, 2019

###### See seminar site

Abstract: See seminar site.

3:00 pm in 341 Altgeld Hall,Friday, September 13, 2019

#### Probabilistic Methods for PDE

###### Kesav Krishnan (UIUC Math)

Abstract: In this talk I will introduce some aspects of Markov Processes, in particular diffusions and their connection to elliptic operators. In particular I will discuss the link between Brownian motion and the Laplacian, the probabilistic interpretation of properties of harmonic functions, such as the mean value theorem, and probabilistic solutions to linear elliptic and parabolic PDE. Time permitting, methods for non-linear PDE will be discussed

4:00 pm in 347 Altgeld Hall,Friday, September 13, 2019

#### Perfect set property

###### Anush Tserunyan   [email] (UIUC Math)

Abstract: A subset of the reals has the perfect set property if it is either countable or contains a copy of Cantor space --- the space of all infinite sequences of 0s and 1s. This is a strong form of the so-called Continuum Hypothesis and the Cantor--Bendixon theorem states that this property holds for all closed subsets of reals. We will discuss this representative theorem of the subject called Descriptive Set Theory and its connection with infinite games, if time permits.

4:00 pm in 141 Altgeld Hall,Friday, September 13, 2019

#### Construction of a Poisson manifold of strong compact type

###### Luka Zwaan (UIUC)

Abstract: I will start the talk with a short introduction to Poisson geometry, going over several equivalent ways of defining a Poisson structure and giving some basic properties and examples. After that I will focus on a specific class of Poisson manifolds, namely those we call ''of compact type''. There are several compactness types, and finding non-trivial examples for the strongest type turns out to be quite difficult. I will sketch a construction which makes use of the many strong properties of K3 surfaces.

Monday, September 16, 2019

3:00 pm in 441 Altgeld Hall,Monday, September 16, 2019

#### Modeling higher algebra with product-and-loop theories

###### William Balderrama (UIUC Math)

Abstract: In this talk, I will introduce the extra homotopical properties of a (suitably infinitary) algebraic theory that make it suitable for modeling spectral, or otherwise higher, algebra, rather than merely derived forms of ordinary algebra. To illustrate the utility of this viewpoint, I will indicate some of the computational tools that can be constructed and understood from this perspective. Time permitting, I will discuss some applications to chromatic homotopy theory.

3:00 pm in 243 Altgeld Hall,Monday, September 16, 2019

#### Packing Lagrangian Tori

###### Ely Kerman (Illinois)

Abstract: While Lagrangian tori bound no volume in dimensions greater than two, in some ways, they behave as if they do. For example, it takes a nontrivial amount of energy to displace one. In this spirit, one might also ask if there are obstructions to embedding many disjoint (integral) Lagrangian tori in a fixed symplectic manifold of finite volume. In this talk I will discuss two results in this direction. The first asserts that the Clifford torus in $S^2 \times S^2$ is a maximal Lagrangian packing in the sense that any other integral Lagrangian torus must intersect it. The second result shows that a natural candidate for a maximal symplectic packing of a polydisc actually fails to be maximal. This is joint work with Richard Hind.

5:00 pm in 241 Altgeld Hall,Monday, September 16, 2019

#### Stability property of conditional probabilities

###### Chris Linden (UIUC)

Abstract: We will present the second part on different forms of equivalences for conditional probabilities developed in the IGL project last semester.

Tuesday, September 17, 2019

12:00 pm in 243 Altgeld Hall,Tuesday, September 17, 2019

#### Coloring invariants of knots and links are often intractable

###### Eric Sampterton (Illinois Math)

Abstract: I’ll give an overview of my result with Greg Kuperberg concerning the computational complexity of G-coloring invariants of knots, where G is a finite, simple group. We have a similar theorem for closed 3-manifolds. I’ll try to give a sense of the commonalities of the two proofs (e.g. “reversible computing with a combinatorial TQFT”), as well as where they differ (there’s some interesting algebraic topology that needed developing in the knot case). Time permitting, I’ll discuss the special case of hyperbolic knots and 3-manifolds.

1:00 pm in 345 Altgeld Hall,Tuesday, September 17, 2019

#### Conjugacy classes of automorphism groups of linearly ordered structures

###### Aleksandra Kwiatkowska (Universität Münster and Uniwersytet Wrocławski)

Abstract: In the talk, we will address the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a countable structure or of a Fraisse limit) that is extremely amenable and has ample generics. In fact, it is unknown if there exists a linearly ordered structure whose automorphism group has a comeager $2$-dimensional diagonal conjugacy class.
We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager 2-dimensional diagonal conjugacy class. Moreover, we provide general conditions implying that there is no comeager conjugacy class or comeager $2$-dimensional diagonal conjugacy class in the automorphism group of an ordered Fraisse limit.
This is joint work with Maciej Malicki.

1:00 pm in 347 Altgeld Hall,Tuesday, September 17, 2019

#### Finite Elements for Curvature

###### Kaibo Hu   [email] (University of Minnesota)

Abstract: We review the elasticity (linearized Calabi) complex, its cohomology and potential applications in differential geometry and continuum defect theory. We construct discrete finite element complexes. In particular, this leads to new finite element discretization for the 2D linearized curvature operator. Compared with classical discrete geometric approaches, e.g., the Regge calculus, the new finite elements are conforming. The construction is based on a Bernstein-Gelfand-Gelfand type diagram chase with various finite element de Rham complexes. This is a joint work with Snorre H. Christiansen.

2:00 pm in 243 Altgeld Hall,Tuesday, September 17, 2019

#### Connected Fair Detachments of Hypergraphs

###### Amin Bahmanian (ISU Math)

Abstract: Let $\mathcal G$ be a hypergraph whose edges are colored. An $(\alpha,n)$-detachment of $\mathcal G$ is a hypergraph obtained by splitting a vertex $\alpha$ into $n$ vertices, say $\alpha_1,\dots,\alpha_n$, and sharing the incident hinges and edges among the subvertices. A detachment is fair if the degree of vertices and multiplicity of edges are shared as evenly as possible among the subvertices within the whole hypergraph as well as within each color class. We find necessary and sufficient conditions under which a $k$-edge-colored hypergraph $\mathcal G$ has a fair detachment in which each color class is connected. Previously, this was not even known for the case when $\mathcal G$ is an arbitrary graph (i.e. 2-uniform hypergraph). We exhibit the usefulness of our theorem by proving a variety of new results on hypergraph decompositions, and completing partial regular combinatorial structures.

3:00 pm in 243 Altgeld Hall,Tuesday, September 17, 2019

#### P=W, a strange identity for Hitchin systems

###### Zili Zhang (U Michigan)

Abstract: Start with a compact Riemann surface X with marked points and a complex reductive group G. According to Hitchin-Simpson’s nonabelian Hodge theory, the pair (X,G) comes with two new complex varieties: the character variety M_B and the Higgs moduli M_D. I will present some aspects of this story and discuss an identity P=W indexed by affine Dynkin diagrams – occurring in the singular cohomology groups of M_D and M_B, where P and W dwell. Based on joint work with Junliang Shen.

Wednesday, September 18, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, September 18, 2019

#### Polish groups whose measure preserving actions are whirly

###### Pavlos Motakis (UIUC Math)

Abstract: Let $\mathrm{MALG}(X)$ denote the measure algebra of a standard probability space $(X,\mu)$. A measure preserving action of a Polish group $G$ on $\mathrm{MALG}(X)$ is called whirly if for any $A, B$ in $\mathrm{MALG}(X)$ with positive measure and for any open neighborhood $U$ of the identity of $G$ there exists $g\in U$ so that $(gA)\cap B$ has positive measure. We follow a paper of Glasner–Tsirelson–Weiss to show that if $G$ is certain type of Polish group, namely a Lévy group, then any non-trivial Borel action on $\mathrm{MALG}(X)$ is whirly. We also show that the Polish group $L_0(\mathbb{T})$ of all measurable functions $[0,1] \to \mathbb{T}$ is Lévy using a suitable concentration inequality.
This is a follow up to the lecture of D. Ihli on 09/11/2019.

4:00 pm in 447 Altgeld Hall,Wednesday, September 18, 2019

#### Intro to the Gorsky-Negut wall-crossing conjecture

###### Josh Wen (Illinois Math)

Abstract: The Hilbert scheme of points on the plane is a space that by now has been connected to many areas outside of algebraic geometry: e.g. algebraic combinatorics, representation theory, knot theory, etc. The equivariant K-theory of these spaces have a few distinguished bases important to making some of these connections. A new entrant to this list of bases is the Maulik-Okounkov K-theoretic stable bases. They depend in a piece-wise constant manner by a real number called the slope, and the numbers where the bases differ are called the walls. Gorsky and Negut have a conjecture relating the transition between bases when the slope crosses a wall to the combinatorics of q-Fock spaces for quantum affine algebras. I'll try to introduce as many of the characters of this story as I can as well as discuss a larger picture wherein these stable bases are geometric shadows of things coming from deformation quantization.

Thursday, September 19, 2019

11:00 am in 241 Altgeld Hall,Thursday, September 19, 2019

#### Indivisibility and divisibility of class numbers of imaginary quadratic fields

###### Olivia Beckwith (Illinois)

Abstract: For any prime p > 3, the strongest lower bounds for the number of imaginary quadratic fields with discriminant down to -X for which the class group has trivial (resp. non-trivial) p-torsion are due to Kohnen and Ono (Soundararajan). I will discuss refinements of these classic results in which we consider the imaginary quadratic fields for which the class number is indivisible (divisible) by p and which satisfy the property that a given finite set of rational primes split in a prescribed way. We prove a lower bound for the number of such fields with discriminant down to -X which is of the same order of magnitude as in Kohnen and Ono's (Soundararajan's) results. For the indivisibility case, we rely on a result of Wiles establishing the existence of imaginary quadratic fields with trivial p-torsion in their class groups which satisfy a finite set of local conditions, and a result of Zagier which says that the Hurwitz class numbers are the Fourier coefficients of a mock modular form.

1:00 pm in 464 Loomis,Thursday, September 19, 2019

#### QUANTUM GRAVITY AND THE SWAMPLAND

###### Gary Shiu (University of Wisconsin)

Abstract: String theory seems to offer an enormous number of possibilities for low energy physics. The huge set of solutions is often known as the String Theory Landscape. In recent years, however, it has become clear that not all quantum field theories can be consistently coupled to gravity. Theories that cannot be ultraviolet completed in quantum gravity are said to be in the Swampland. In this talk, I’ll discuss some conjectured properties of quantum gravity, evidences for them, and their applications to cosmology.

2:00 pm in 243 Altgeld Hall,Thursday, September 19, 2019

#### Inductive limits of C*-algebras and compact quantum metric spaces

###### Konrad Aguilar (Arizona State University)

Abstract: In this talk, we will place quantum metrics, in the sense of Rieffel, on certain unital inductive limits of C*-algebras built from quantum metrics on the terms of the given inductive sequence with certain compatibility conditions. One of these conditions is that the inductive sequence forms a Cauchy sequence of quantum metric spaces in the dual Gromov-Hausdorff propinquity of Latremoliere. Since the dual propinquity is complete, this will produce a limit quantum metric space. Based on our assumptions, we then show that the C*-algebra of this limit quantum metric space is isomorphic to the given inductive limit, which finally places a quantum metric on the inductive limit. This then immediately allows us to establish a metric convergence of the inductive sequence to the inductive limit. Another consequence to our construction is that we place new quantum metrics on all unital AF algebras that extend our previous work with Latremoliere on unital AF algebras with faithful tracial state.

2:00 pm in 347 Altgeld Hall,Thursday, September 19, 2019

#### The Yang Mills Problem for Probabilists

###### Kesav Krishnan (UIUC math)

Abstract: We aim to introduce the problem of rigorously defining the Yang-Mills field from the probability perspective. In this first talk, we will introduce lattice guage theory, and some geometric preliminaries

4:00 pm in 245 Altgeld Hall,Thursday, September 19, 2019

#### On the container method

###### Jozsef Balogh   [email] (University of Illinois at Urbana-Champaign)

Abstract: We will give a gentle introduction to a recently-developed technique, The Container Method’, for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of 'containers' for the independent sets, each of which contains few edges. The container method is very useful counting discrete structures with certain properties; transferring theorems into random environment; and proving the existence discrete structures satisfying some important properties. In the first half of the talk we will attempt to convey a general high-level overview of the method, in particular how independent sets in hypergraphs could be used to model various problems in combinatorics; in the second, we will describe a few illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry.

Friday, September 20, 2019

2:00 pm in 147 Altgeld Hall,Friday, September 20, 2019

#### Escaping nontangentiality: Towards a controlled tangential amortized Julia-Carathéodory theory

###### Meredith Sargent (University of Arkansas)

Abstract: Let $f: D \rightarrow \Omega$ be a complex analytic function. The Julia quotient is given by the ratio between the distance of $f(z)$ to the boundary of $\Omega$ and the distance of $z$ to the boundary of $D.$ A classical Julia-Carathéodory type theorem states that if there is a sequence tending to $\tau$ in the boundary of $D$ along which the Julia quotient is bounded, then the function $f$ can be extended to $\tau$ such that $f$ is nontangentially continuous and differentiable at $\tau$ and $f(\tau)$ is in the boundary of $\Omega.$ We develop an extended theory when $D$ and $\Omega$ are taken to be the upper half plane which corresponds to amortized boundedness of the Julia quotient on sets of controlled tangential approach, so-called $\lambda$-Stolz regions, and higher order regularity, including but not limited to higher order differentiability, which we measure using $\gamma$-regularity. I will discuss the proof, along with some applications, including moment theory and the fractional Laplacian. This is joint work with J.E. Pascoe and Ryan Tully-Doyle.

2:00 pm in 347 Altgeld Hall,Friday, September 20, 2019

#### Hopf Ore Extensions

###### Hongdi Huang (University of Waterloo (visiting UIUC for F19))

Abstract: Brown, O'Hagan, Zhang, and Zhuang gave a set of conditions on an automorphism $\sigma$ and a $\sigma$-derivation $\delta$ of a Hopf $k$-algebra $R$ for when the skew polynomial extension $T=R[x, \sigma, \delta]$ of $R$ admits a Hopf algebra structure that is compatible with that of $R$. In fact, they gave a complete characterization of which $\sigma$ and $\delta$ can occur under the hypothesis that $\Delta(x)=a\otimes x +x\otimes b +v(x\otimes x) +w$, with $a, b\in R$ and $v, w\in R\otimes_k R$, where $\Delta: R\to R\otimes_k R$ is the comultiplication map. In this paper, we show that after a change of variables one can in fact assume that $\Delta(x)=\beta^{-1}\otimes x +x\otimes 1 +w$, with $\beta$ is a grouplike element in $R$ and $w\in R\otimes_k R,$ when $R\otimes_k R$ is a domain and $R$ is noetherian. In particular, this completely characterizes skew polynomial extensions of a Hopf algebra that admit a Hopf structure extending that of the ring of coefficients under these hypotheses. We show that the hypotheses hold for domains $R$ that are noetherian cocommutative Hopf algebras of finite Gelfand-Kirillov dimension.

3:00 pm in 341 Altgeld Hall,Friday, September 20, 2019

#### Introduction to Quasiconformal and Quasisymmetric maps on metric spaces

###### Stathis Chrontsios (UIUC Math)

Abstract: The talk will be a quick introduction to quasiconformal and quasisymmetric maps on metric spaces. I will start by describing how quasiconformal maps first appeared as generalizations of conformal maps on the complex plane and how they were generalized in arbitrary metric spaces. In addition, I will present how they gave rise to quasisymmetric maps on the real line and their later generalization in metric spaces. Moreover, I will discuss interesting quasisymmetric invariants and the definition of the conformal gauge. Last but not least, I will mention some applications this theory has had in Geometric Group Theory and some open problems.

3:00 pm in 1 Illini Hall,Friday, September 20, 2019

#### The prime number theorem through the Ingham-Karamata Tauberian theorem

###### Gregory Debruyne (Illinois Math)

Abstract: It is well-known that the prime number theorem can be deduced from certain Tauberian theorems. In this talk, we shall present a Tauberian approach that is perhaps not that well-known through the Ingham-Karamata theorem. Moreover, we will give a recently discovered "simple" proof of a so-called one-sided version of this theorem. We will also discuss some recent developments related to the Ingham-Karamata theorem. The talk is based on work in collaboration with Jasson Vindas.

4:00 pm in 345 Altgeld Hall,Friday, September 20, 2019

#### A Logician's Introduction to the Problem of P vs. NP

###### Alexi Block Gorman (UIUC Math)

Abstract: Central to much of computer science, and some areas of mathematics, are questions about various problems' computability and complexity (whether the problem can be solved "algorithmically," and how "hard" it is to do so). In this talk, I will first give an overview of the complexity hierarchy for machines (from finite automata to Turing machines) and the mathematical properties of the space of languages that we associate with them. Next, I will discuss the relationship of deterministic and non-deterministic machines, which will allow us to segue from questions of computability to that of complexity. Finally, I will give a precise formulation of the problem of P vs. NP, and try to illustrate why the problem remains rather elusive. This talk does not require any background in logic or computer science, and should be accessible to all graduate students.

4:00 pm in 347 Altgeld Hall,Friday, September 20, 2019

#### Classifying Homomorphisms from the Braid and Symmetric Groups

###### Alice Chudnovsky (UIUC Math)

Abstract: One goal of group theory is to understand all possible homomorphisms between two groups. My group characterized maps from the symmetric group and braid group to solvable groups, abelian groups, dihedral groups, free groups, and other groups. This process was facilitated by the concept of Totally Symmetric Sets (TSS), a “basis” of sorts for a group, developed by Margalit, Kordek and Chen. TSS induces a correspondence: given a group homomorphism φ: G → H and a totally symmetric set S of size n in G, φ(S) is either a totally symmetric set of size n in H or of size 1. By determining two groups’ totally symmetric sets, we were able to prove for many cases that all possible homomorphisms factor through a cyclic group. Moreover, for those groups, we drew conclusions on the structure of their totally symmetric sets, and gave upper bounds for their possible size.

4:00 pm in 141 Altgeld Hall,Friday, September 20, 2019

#### A Geometric Proof of Lie's Third Theorem

###### Shuyu Xiao (UIUC)

Abstract: There are three basic results in Lie theory known as Lie's three theorems. These theorems together tell us that: up to isomorphism, there is a one-to-one correspondence between finite-dimensional Lie algebras and simply connected Lie groups. While the first two theorems are easy to prove with the most basic differential geometry knowledge, the third one is somehow a deeper result which needs relatively advanced tools. In this talk, I will go over the proof given by Van Est, in which he identifies any finite-dimensional Lie algebra with a semi-direct product of its center and its adjoint Lie algebra. I will introduce Lie group cohomology, Lie algebra cohomology and how they classify the abelian extensions of Lie groups and Lie algebras and thus determine the Lie algebra structure on the semi-direct product.

Monday, September 23, 2019

3:00 pm in 243 Altgeld Hall,Monday, September 23, 2019

#### Moment maps in quantum mechanics

###### Eugene Lerman (Illinois)

Abstract: I will try to explain how Atiyah-Guillemin-Sternberg-Kirwan convexity theorem shows up in quantum mechanics. The talk is expository. I claim no originality.

3:00 pm in 441 Altgeld Hall,Monday, September 23, 2019

#### Splitting $BP\langle 1\rangle \wedge BP\langle 1\rangle$ at odd primes

###### Liz Tatum (UIUC Math)

Abstract: The Adams Spectral Sequence is a tool for approximating $\pi_{*}X$, where $X$ is a connective spectrum. If $E$ is a ring spectrum satisfying certain properties, then we can define an $E$-based Adams spectral sequence converging to $\pi_{*}\hat{X}$, where $\hat{X}$ is the $E$-completion of $X$. When $E_{*}E$ is flat over $E_{*}$, the $E^{2}$-page of the spectral sequence can be described as $Ext_{E_{*}E}(E_{*}, E_{*}X)$. But if $E_{*}E$ is not flat over $E_{*}$, then there is no such description. Instead, we must study $E\wedge E$ to understand the spectral sequence. The Brown-Peterson spectra $BP\langle n \rangle$ are an example of such spectra. One approach is to split the product $E \wedge E$ into more manageable pieces. When $n=1$, we can construct a splitting $BP\langle 1 \rangle \wedge BP\langle 1 \rangle$ as $\vee_{k=0}^{\infty}\Sigma^{2k(p-1)} BP\langle 1 \rangle 1 \wedge B(k)$, where $B(k)$ is the $k^{th}$ integral Brown-Gitler spectrum. We give a sketch of Kane's construction of this splitting for odd primes.

5:00 pm in 241 Altgeld Hall,Monday, September 23, 2019

#### Boundary Representations

###### Chris Linden (UIUC)

Abstract: We will continue the discussion of boundary of previously considered operator algebras, and launch into Arveson’s general general.

Tuesday, September 24, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, September 24, 2019

#### Speeds of hereditary properties and mutual algebricity

###### Caroline Terry (U Chicago Math)

Abstract: A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property $H$, the speed of $H$ is the function which sends an integer $n$ to the number of distinct elements in $H$ with underlying set $\{1,...,n\}$. Not just any function can occur as the speed of hereditary graph property. Specifically, there are discrete "jumps" in the possible speeds. Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollobás, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized. In contrast to this, many aspects of this problem in the hypergraph setting remained unknown. In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds. The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss. This is joint work with Chris Laskowski.

2:00 pm in 347 Altgeld Hall,Tuesday, September 24, 2019

#### On the potential theory of Markov processes with jump kernels decaying at the boundary

###### Zoran Vondracek (University of Zagreb)

Abstract: Consider a $\beta$-stable process in the Euclidean space $\mathbb{R}^d$, $0<\beta\le 2$, which is killed upon exiting an open subset $D$. The killed process is then subordinated via an independent $\gamma$-stable subordinator. The resulting process $Y^D$ is called a subordinate killed stable process. In two recent papers, it has been shown that the potential theory of this process exhibits some interesting features. The first one is the form of the jumping kernel which depends on the distance of points to the boundary in a novel way. The second and unexpected feature is the fact that for some values of the stability index $\gamma$, the boundary Harnack principle fails. In the first part of the talk, I will review these results. The second part of the talk will be devoted to ongoing work on potential theory of jump processes in open subset $D$ of $\mathbb{R}^d$ defined through their jumping kernels that depend not only on the distance between two points, but also on the distance of each point to the boundary $\partial D$ of the state space $D$. Joint work with Panki Kim and Renming Song

2:00 pm in 243 Altgeld Hall,Tuesday, September 24, 2019

#### Defective DP-colorings of sparse multigraphs

###### Fuhong Ma (Shangdong University)

Abstract: DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle. We introduce and study defective DP-colorings of graphs and multigraphs with 2 colors. Each vertex $v$ of a multigraph $G$ has colors $\alpha(v)$ and $\beta(v)$ in its list. In an $(i,j)$-coloring of $G$, if $v$ is colored with $\alpha(v)$, then it can be incident to at most $i$ 'conflicting' edges, otherwise it can be incident to at most $j$ such edges. We concentrate on $(i,j)$-colorings of sparse multigraphs.

Let $f_{DP}(i,j,n)$ be the minimum number of edges that may have an $n$-vertex $(i,j)$-critical multigraph, that is, a multigraph $G$ that has no $(i,j)$-defective DP-coloring but whose every proper subgraph has such a coloring. For all $i$ and $j$, we find linear lower bounds on $f_{DP}(i,j,n)$ that are exact for infinitely many $n$.

3:00 pm in 243 Altgeld Hall,Tuesday, September 24, 2019

#### Motivic Chern classes and Iwahori invariants of principal series

###### Changjian Su (University of Toronto)

Abstract: Let G be a split reductive p-adic group. In the Iwahori invariants of an unramified principal series representation of G, there are two bases, one of which is the so-called Casselman basis. In this talk, we will prove conjectures of Bump, Nakasuji and Naruse about certain transition matrix coefficients between these two bases. The ingredients of the proof involve Maulik and Okounkov's stable envelopes and Brasselet--Schurmann--Yokura's motivic Chern classes for the complex Langlands dual groups. This is based on joint work with P. Aluffi, L. Mihalcea and J. Schurmann.

4:00 pm in 245 Altgeld Hall,Tuesday, September 24, 2019

#### The University of Illinois Math Community in Prison: Calculus, Coding, and Beyond

###### Simone Sisneros-Thiry   [email] (University of Illinois at Urbana-Champaign)

Abstract: From spring 2018 through spring 2019, a cohort of Education Justice Project (EJP) students enrolled in the University of Illinois calculus series (Math 115-231) at the Danville Correctional Center. The value of and need for math education in prisons has been keenly felt by EJP students. Our interest in mathematics and related fields has prompted these courses as a part of an increasingly robust math and engineering curriculum. Our presentation will introduce the history and current status of EJP math programming. It will focus on student motivation, interest, and background in mathematics, and will include reflections by students and instructors on approaches and outcomes.

Wednesday, September 25, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, September 25, 2019

#### Spatial actions of Lévy groups (90 minute talk)

###### Pavlos Motakis (UIUC Math)

Abstract: Following up last week's seminar we prove a theorem of Glasner–Tsirelson–Weiss according to which any spacial action of a Lévy group is trivial.

4:00 pm in 447 Altgeld Hall,Wednesday, September 25, 2019

#### The renormalized De Rham functor

###### Ciaran O'Neill (Illinois Math)

Abstract: I’ll start with some background, then give the definition of the renormalized De Rham functor (as defined by Drinfeld and Gaitsgory). This comes with a natural transformation to the ordinary De Rham functor. I’ll mention how this can potentially be used to prove Kirwan surjectivity in certain circumstances. There will also be an example or two.

4:00 pm in 245 Altgeld Hall,Wednesday, September 25, 2019

#### Tackling Gender Inequality in Math

###### Teryl Brewster (Office of Inclusion and Intercultural Relations)

Abstract: Professionals from the Office of Inclusion and Intercultural Relations will present a workshop focused on understanding the difference between sex and gender, how people face stereotypes because of the gender binary, and increasing awareness of where gender biases come from. The workshop will highlight these issues in the academic math context.

Thursday, September 26, 2019

11:00 am in 241 Altgeld Hall,Thursday, September 26, 2019

#### Large prime gaps and Siegel zeros

###### Kevin Ford (Illinois Math)

Abstract: We show that the existence of zeros of Dirichlet L-functions very close to 1 ("Siegel zeros") implies larger prime gaps than are currently known. We also present a heuristic argument that the existence of Siegel zeros implies gaps of larger order than $\log^2 x$, that is, larger than the Cramer conjecture.

12:00 pm in 243 Altgeld Hall,Thursday, September 26, 2019

#### Finite Rigid Sets in Arc Complexes

###### Emily Shinkle (Illinois Math)

Abstract: The arc complex is a simplicial complex associated to a surface. In this talk, I will describe my recent result about the existence of finite rigid sets in arc complexes: finite simplicial subcomplexes such that any injection into the arc complex of another surface with arc complex of the same dimension is induced by a homeomorphism of the associated surfaces. I will give a brief overview of my proof and also discuss some related results.

1:00 pm in 464 Loomis,Thursday, September 26, 2019

#### Holographic entanglement and quantum gravity in finite regions

###### Will Donnelly (Perimeter Institute)

Abstract: The T\bar{T} deformation of two-dimensional holographic conformal field theory is conjectured to be dual to quantum gravity in a finite region of three-dimensional anti-de Sitter spacetime. We study entanglement entropy in this theory and its relation to quantum fluctuations of the dual geometry. We derive the correspondence between the T\bar{T} flow equation and the Wheeler-DeWitt equation with a negative cosmological constant. By fixing the resulting emergent diffeomorphism symmetry, we obtain a differential equation for the sphere partition function which can be solved exactly. The solution can be expressed as a Euclidean path integral along a particular complex contour. We then apply this result to study entanglement entropy of the boundary theory for an entangling surface consisting of two antipodal points on the sphere. The entanglement entropy gives the Ryu-Takayanagi formula for a geodesic in a finite region plus quantum corrections. We suggest an interpretation of the latter as entropy of fluctuations of the bulk geodesic length, in accordance with the proposal of Faulkner, Lewkowycz, and Maldacena.

3:00 pm in 347 Altgeld Hall,Thursday, September 26, 2019

#### Ehrhart theory of Newton polytopes in algebraic combinatorics

###### McCabe Olsen   [email] (The Ohio State University)

Abstract: Motivated by the work of Monical, Tokcan, and Yong on the saturated Newton polytope (SNP) in algebraic combinatorics, we study Ehrhart theoretic properties of theses polytopes. We focus primarily upon classifying when these polytopes are Gorenstein and when they have the integer decomposition property. Often, the existence of SNP for these polynomials allows for short combinatorial proofs of the integer decomposition property. For this talk, we will mostly consider the Newton polytopes for Schur polynomials and symmetric Grothendieck polynomials. This is ongoing work which originated at the 2019 Graduate Research Workshop in Combinatorics and is joint with Margaret Bayer, Bennet Goeckner, Suji Hong, Tyrrell McAllister, Casey Pinckney, Julianne Vega, and Martha Yip.

4:00 pm in 245 Altgeld Hall,Thursday, September 26, 2019

#### Fall Department Faculty Meeting

Abstract: The Fall Department Faculty Meeting will be held at 4 p.m. in 245 Altgeld Hall, followed by a reception in 239 Altgeld Hall.

Friday, September 27, 2019

2:00 pm in 347 Altgeld Hall,Friday, September 27, 2019

###### See seminar site

Abstract: See seminar site.

3:00 pm in 341 Altgeld Hall,Friday, September 27, 2019

#### Iwaniec's conjecture

###### Terence Lee John Harris (UIUC Math)

Abstract: The Beurling-Ahlfors transform is given by $(Sf)(z) = \frac{-1}{\pi} \int_{\mathbb{C}} \frac{f(\zeta) }{(z-\zeta)^2} \, d\zeta$. Finding the exact norm of $S$ on $L^p(\mathbb{C})$ is an open problem for $p \neq 2$. In this talk, I will outline some of the basic connections between this problem and other areas, such as martingales, quasiconformal mappings and the calculus of variations.

4:00 pm in 141 Altgeld Hall,Friday, September 27, 2019

#### Thurston’s Construction of pseudo-Anosovs

###### Christopher Loa (UIUC)

Abstract: In the 1970’s, Thurston classified Mod(S) for higher genus surfaces in a widely circulated preprint, “remarkable for its brevity and richness.” This classification turns out to be a trichotomy (finite order, reducible, or pseudo-Anosov), just like the classification of automorphisms of the torus (finite order, reducible, or Anosov). The aim of this talk is to spell out his construction “for a large class of examples of diffeomorphisms in canonical form.” The real treasure of this construction is that it allows us to easily get our hands on pseudo-Anosov maps, a seemingly difficult task. As Thurston himself wrote “. . . it is pleasant to see something of this abstract origin made very concrete.” We motivate the construction by first classifying the automorphisms of the torus. Knowledge of basic linear algebra and hyperbolic geometry is assumed, and familiarity with mapping class groups will be helpful for following along.

4:00 pm in 347 Altgeld Hall,Friday, September 27, 2019

#### Set Tournament

###### MATRIX (UIUC Math)

4:00 pm in 345 Altgeld Hall,Friday, September 27, 2019

#### O-minimal complex analysis according to Peterzil–Starchenko (Part 1)

###### Lou van den Dries (UIUC)

Abstract: This is the first of two survey talks on the subject of the title. Neer (and others?) will follow up with a more detailed treatment in later talks. O-minimal complex analysis is one way that ideas from o-minimality have been used in recent work in arithmetic algebraic geometry (Pila, Zannier, Tsimerman, Klingler,…), the other one being the Pila–Wilkie theorem. The two topics relate because important objects like the family of Weierstrass p-functions turn out to be "o-minimal".

Monday, September 30, 2019

3:00 pm in 243 Altgeld Hall,Monday, September 30, 2019

#### New action-angle variables on coadjoint orbits

###### Yanpeng Li (University of Geneva)

Abstract: The problem of constructing global action-angle variables on coadjoint orbits of compact Lie groups is one of the interesting questions in the theory of integrable systems. A fundamental contribution was made by Guillemin-Sternberg who constructed the Gelfand-Zeitlin integrable systems on coadjoint orbits of the groups SU(n) and SO(n). Recently, toric degeneration techniques allowed for the construction of global action-angle variables on rational coadjoint orbits of compact Lie groups of all types. In this talk, I will present a new approach which aims at constructing global action-angle coordinates on all regular coadjoint orbits of compact Lie groups and on a large family of related Hamiltonian spaces. It combines the results of Ginzburg-Weinstein on the theory of Poisson-Lie groups and the theory of cluster algebras using the "partial tropicalization'' procedure.

3:00 pm in 441 Altgeld Hall,Monday, September 30, 2019

#### The Recognition Principle for Infinite Loop Spaces

###### Brian Shin (UIUC Math)

Abstract: In this expository talk, I'd like to discuss the infinite loop space recognition principle. In particular, I'd like to examine Boardman-Vogt's infinite loop space machine from a modern view point.

4:00 pm in 241 Altgeld Hall,Monday, September 30, 2019

#### Anxiety and Stress Management Workshop

###### Counselling Center Professionals

5:00 pm in 243 Altgeld Hall,Monday, September 30, 2019

#### Anxiety and Stress Management Graduate Student Panel

###### (AWM)

Abstract: Graduate students will share their experiences on how they dealt with the inevitable stress of graduate life, especially during their first year. Also, cookies will be provided!

Tuesday, October 1, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, October 1, 2019

#### Continuity of Functions on the Powerset of the First Uncountable Cardinal

###### William Chan (University of North Texas)

Abstract: Assume the axiom of determinacy. In this talk, we will discuss an almost everywhere uniformization for club subsets of $\omega_1$. Using this uniformization, we will show that every function $\Phi : [\omega_1]^{\omega_1} \rightarrow \omega_1$ has a club $C \subseteq \omega_1$ so that $\Phi | [C]^{\omega_1}$ is a continuous function. This continuity result can be used to make the following cardinality distinction, $|[\omega_1]^{<\omega_1}| < |[\omega_1]^{\omega_1}|$. This is joint work with Stephen Jackson.

2:00 pm in 347 Altgeld Hall,Tuesday, October 1, 2019

#### The genus of generalized sparse random graphs

###### Yifan Jing (UIUC)

Abstract: Determining the genus of graphs is one of the central problems in topological graph theory. In particular, it was proved by Thomassen that determining the genus of any given graph is NP-Complete. Recently, we provided a polynomial-time approximation scheme for the genus of dense graphs. One of the key ingredients there is that we approximate the genus of dense multipartite quasirandom graphs. Motivated by that project, we study the genus of generalized sparse random graphs, that is graphs can be decomposed by finite sum of sparse random graphs. This is joint work with Bojan Mohar.

2:00 pm in 243 Altgeld Hall,Tuesday, October 1, 2019

#### Long monochromatic paths and cycles in $2$-edge-colored graphs with large minimum degree

###### Mikhail Lavrov (Illinois Math)

Abstract: Our main result is a proof for sufficiently large $n$ of the conjecture by Benevides, Łuczak, Scott, Skokan and White that for every positive integer $n$ of the form $n=3t+r$ where $r \in \{0,1,2\}$ and every $n$-vertex graph $G$ with $\delta(G) \ge 3n/4$, in each $2$-edge-coloring of $G$ there exists a monochromatic cycle of length at least $2t+r$.

Our result also implies the conjecture of Schelp that for every sufficiently large $n$, every $(3n-1)$-vertex graph $G$ with minimum degree larger than $3|V(G)|/4$, in every $2$-edge-coloring of $G$, there is a monochromatic path with $2n$ vertices. Joint work with József Balogh, Alexandr Kostochka and Xujun Liu.

3:00 pm in 243 Altgeld Hall,Tuesday, October 1, 2019

#### Standard Conjecture D for matrix factorizations

Abstract: In 1968, Grothendieck posed a family of conjectures concerning algebraic cycles called the Standard Conjectures. The conjectures have been proven in some special cases, but they remain open in general. In 2011, Marcolli-Tabuada realized two of these conjectures as special cases of more general conjectures, involving differential graded categories, which they call Noncommutative Standard Conjectures C and D. The goal of this talk is to discuss a proof, joint with Mark Walker, of Noncommutative Standard Conjecture D in a special case which does not fall under the purview of Grothendieck's original conjectures: namely, in the setting of matrix factorizations.

4:00 pm in 245Altgeld Hall,Tuesday, October 1, 2019

#### Limitations on All Known (and Some Unknown) Approaches to Matrix Multiplication

###### Virginia Vassilevska-Williams   [email] (Massachusetts Institute of Technology)

Abstract: In this talk we will consider the known techniques for designing Matrix Multiplication algorithms. The two main approaches are the Laser method of Strassen, and the Group theoretic approach of Cohn and Umans. We define generalizations that subsume these two approaches: the Galactic and the Universal method; the latter is the most general method there is. We then design a suite of techniques for proving lower bounds on the value of $\omega$, the exponent of matrix multiplication, which can be achieved by algorithms using many tensors $T$ and the Galactic method. Some of our techniques exploit local' properties of $T$, like finding a sub-tensor of $T$ which is so weak' that $T$ itself couldn't be used to achieve a good bound on $\omega$, while others exploit global' properties, like $T$ being a monomial degeneration of the structural tensor of a group algebra. The main result is that there is a universal constant $\ell>2$ such that a large class of tensors generalizing the Coppersmith-Winograd tensor $CW_q$ cannot be used within the Galactic method to show a bound on $\omega$ better than $\ell$, for any $q$. We give evidence that previous lower-bounding techniques were not strong enough to show this. The talk is based on joint work with Josh Alman, which appeared in FOCS 2018. More recently, Alman showed how to extend our techniques so that they apply to the Universal method as well. In particular, Alman shows that the Coppersmith-Winograd tensor cannot yield a better bound on $\omega$ than 2.16805 even using the Universal method.

Wednesday, October 2, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, October 2, 2019

#### Generic measure-preserving transformations are whirly (50 minute talk)

###### Dakota Ihli (UIUC Math)

Abstract: We show that the action of a generic measure-preserving transformation on a standard probability space is whirly. This will be the final part in our series of seminar talks discussing the paper "Spatial and non-spatial actions of Polish groups" by Glasner and Weiss.

4:00 pm in 447 Altgeld Hall,Wednesday, October 2, 2019

#### Intro to Gromov-Witten invariants

Abstract: Gromov-Witten invariants (often) count the number of curves (= Riemann surfaces) of a fixed genus in a projective variety (= nice complex manifold). I will introduce these invariants and compute a few examples.

Thursday, October 3, 2019

11:00 am in 241 Altgeld Hall,Thursday, October 3, 2019

#### Optimality of Tauberian theorems

###### Gregory Debruyne (Illinois & Ghent University)

Abstract: The Wiener-Ikehara and Ingham-Karamata theorems are two celebrated Tauberian theorems which are known to lead to short proofs of the prime number theorem. In this talk, we shall investigate quantified versions of these theorems and show that these are optimal. For the optimality, rather than constructing counterexamples, we shall use an attractive functional analysis argument based on the open mapping theorem. The talk is based on work in collaboration with David Seifert and Jasson Vindas.

12:00 pm in 243 Altgeld Hall,Thursday, October 3, 2019

#### Constructing pseudo-Anosov homeomorphisms using positive twists

###### Yvon Verberne (U Toronto)

Abstract: Thurston obtained the first construction of pseudo-Anosov homeomorphisms by showing the product of two parabolic elements is pseudo-Anosov. A related construction by Penner involved constructing whole semigroups of pseudo-Anosov homeomorphisms by taking appropriate compositions of Dehn twists along certain families of curves. In this talk, I will present a new construction of pseudo-Anosov homeomorphisms and discuss some of the unique properties associated to maps obtained through this new construction.

1:00 pm in 464 Loomis Laboratory ,Thursday, October 3, 2019

#### The power of string theory in TTbar and related theories

###### David Kutasov (University of Chicago Physics)

Abstract: I describe the recent discovery of a class of non-local field theories that can be thought of as irrelevant deformations of two dimensional conformal field theories, focusing on their description as deformations of string theory on three dimensional anti-de-Sitter space.

2:00 pm in 347 Altgeld Hall,Thursday, October 3, 2019

#### Introduction to Spin Glasses Part I

###### Qiang Wu (UIUC Math)

Abstract: I will introduce some spin glass models in the first talk, such as Curie-Weiss(CW) model, Sherrington-Kirkpatrick(SK) model etc, then particularly discuss the high temperature analysis of SK model by Guerra’s interpolation.

3:00 pm in 347 Altgeld Hall,Thursday, October 3, 2019

#### Plabic R-Matrices

###### Sunita Chepuri   [email] (University of Minnesota, Twin Cities)

Abstract: Postnikov's plabic graphs in a disk are used to parametrize totally nonnegative Grassmannians. One of the key features of this theory is that if a plabic graph is reduced, the face weights can be uniquely recovered from boundary measurements. On surfaces more complicated than a disk this property is lost. In this talk, we investigate a certain semi-local transformation of weights for plabic networks on a cylinder that preserves boundary measurements. We call this a plabic R-matrix. We explore the properties of the plabic R-matrix, including the symmetric group action it induces on plabic networks and its underlying cluster algebra structure.

4:00 pm in 245 Altgeld Hall,Thursday, October 3, 2019

#### The Rise of Peer-to-Peer Insurance and Its Mathematical Modeling

###### Runhuan Feng   [email] (University of Illinois at Urbana-Champaign)

Abstract: Peer-to-peer (P2P) insurance is a decentralized network in which participants pool their resources together to compensate those who suffer losses. It is a revival of a centuries-old practice in many ancient societies where members care for each other’s financial needs in the event of misfortune. With the aid of internet technology, P2P insurance is becoming a transparent, high-tech and low-cost alternative to traditional insurance and is viewed by many as a huge disruptor to the traditional insurance industry in the same way Uber is to the taxi industry. P2P insurance took an unexpected twist in the Chinese market. A new business model called “mutual aid” is largely driven by many non-insurance tech firms, including the e-commerce giant, Alibaba. In less than three years, the mutual aid industry amassed close to 260 million participants, which is nearly 20% of the Chinese population, or equivalently, 80% of the US population. Such an unprecedented development is causing huge anxiety for insurers and regulators in China and being closely watched by their peers around the world. Despite the fast-changing landscape in this field, there has no previous academic literature for the theoretical underpinning of P2P insurance. Our research team presents the first such effort to build the mathematical framework under which the design, engineering and management of mutual aid and P2P insurance can be studied. This presentation is based on joint work with Samal Abdikerimova and Chongda Liu.

Friday, October 4, 2019

2:00 pm in 347 Altgeld Hall,Friday, October 4, 2019

###### See seminar site

Abstract: See seminar site.

3:00 pm in 341 Altgeld Hall,Friday, October 4, 2019

#### A sharp eigenvalue inequality for signed graph Laplacians

###### Derek Kielty (UIUC Math)

Abstract: Abstract: The collection of eigenvalues of a graph Laplacian matrix carries information about the topology the graph. Graph Laplacians are also closely related to discretizations of the Laplacian differential operator. Signed graph Laplacians are a generalization that encode attractions or repulsions between the vertices of a graph by assigning weights to the edges of the graph. In this talk I give an introduction to graph Laplacians and will discuss a sharp inequality on the eigenvalues of signed graph Laplacians (based on joint work with Ikemefuna Agbanusi and Jared Bronski).

4:00 pm in 347 Altgeld Hall,Friday, October 4, 2019

#### Error Correcting Codes

###### Aubrey Laskowski   [email] (UIUC Math)

Abstract: Error correcting codes (ECCs) are some of the core connecting components between coding theory and information theory, and much of modern communication involves some aspect of an ECC - from CDs to Amazon Web Services and even to space missions. ECCs provide the ability to decode information from noise channels by providing a small amount of redundant data. I will discuss the history of ECCs, as well as specific codes such as Hamming codes and Reed Solomon codes. Important parameters of ECCs as a historical perspective will be discussed and, time permitting, I will introduce how ECCs connect to machine learning. For the history, no specific background is assumed, but linear and abstract algebra will likely be usful for other aspects of the talk.

4:00 pm in 345 Altgeld Hall,Friday, October 4, 2019

#### O-minimal complex analysis according to Peterzil–Starchenko (Part 2)

###### Lou van den Dries (UIUC)

Abstract: This is the first of two survey talks on the subject of the title. Neer (and others?) will follow up with a more detailed treatment in later talks. O-minimal complex analysis is one way that ideas from o-minimality have been used in recent work in arithmetic algebraic geometry (Pila, Zannier, Tsimerman, Klingler,…), the other one being the Pila–Wilkie theorem. The two topics relate because important objects like the family of Weierstrass p-functions turn out to be "o-minimal".

4:00 pm in TBD,Friday, October 4, 2019

#### Graduate Research Opportunities for Women

Abstract: The GROW 2019 conference is aimed at female-identified undergraduate students who may be interested in pursuing a graduate degree in mathematics. The conference is open to undergraduates from all around the U.S.

Saturday, October 5, 2019

9:00 am in TBD,Saturday, October 5, 2019

#### Graduate Research Opportunities for Women

Abstract: The GROW 2019 conference is aimed at female-identified undergraduate students who may be interested in pursuing a graduate degree in mathematics. The conference is open to undergraduates from all around the U.S.

Sunday, October 6, 2019

9:00 am in TBD,Sunday, October 6, 2019

#### Graduate Research Opportunities for Women

Abstract: The GROW 2019 conference is aimed at female-identified undergraduate students who may be interested in pursuing a graduate degree in mathematics. The conference is open to undergraduates from all around the U.S.

Monday, October 7, 2019

3:00 pm in 441 Altgeld Hall,Monday, October 7, 2019

#### Motivating Higher Toposes: Geometric Characteristic Classes

###### Joseph Rennie (UIUC Math)

Abstract: This will be part one of two talks aimed at motivating higher topos theory from physics. In this talk we will give a brief history of classical obstructions for manifolds, then suddenly find ourselves naturally requiring tools from higher topos theory. In the end, we shall see how working with simplicial sheaves on Manifolds allows us to define (but more importantly compute) geometric characteristic classes.

5:00 pm in 241 Altgeld Hall,Monday, October 7, 2019

#### Some properties of alpha fidelities.

###### Xiaojing Yan

Abstract: The alpha fidelities constitute a family of information measures that generalize the well-known fidelity, inheriting many of its properties. In the first half of the talk, we will introduce some natural properties for quantum alpha fidelities, including data-processing inequalities and monotonicity and so on. In the second, we will calculate the quantum α-fidelity between unitary orbits.

Tuesday, October 8, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, October 8, 2019

#### A Categorical Semantics for Linear (and Quantum) Dependent Type Theory

###### Kohei Kishida (UIUC Philosophy)

Abstract: Desirable features of a quantum programming language include: treating both quantum resources and classical control; being a functional language admitting semantics and other formal methods; and dependent types. The type theory of such a language must be linear to reflect the linearity of quantum processes, and we want it to involve parameters (values that are known at circuit generation time) and states (values known at circuit execution time) to describe and generate quantum circuits. The goal of this talk is to provide a general semantic structure for linear dependent type theory of that sort. We review categorical models of quantum processes and the sets-and-functions model of classical dependent type theory, and show how they can be integrated to model linear dependent type theory of classical parameters and quantum states. This is joint work with Frank Fu, Julien Ross, and Peter Selinger.

2:00 pm in 347 Altgeld Hall,Tuesday, October 8, 2019

#### Introduction to Spin Glasses Part II

###### Qiang Wu (UIUC Math)

Abstract: This time we will discuss the parisi formula of free energy, I will describe how to derive the formula along with the parisi PDE. If time permits, ultrametricity of asymptotic Gibbs measure will be briefly introduced from probabilistic and geometric view.

2:00 pm in 243 Altgeld Hall,Tuesday, October 8, 2019

#### Problems and results on $1$-cross intersecting set pair systems

###### Zoltán Füredi (Rényi Institute, Budapest, Hungary)

Abstract: The notion of cross intersecting set pair system of size $m$, $(\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m)$ with $A_i \cap B_i = \emptyset$ and $A_i \cap B_j \ne \emptyset$, was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that $m \le {{a+b} \choose a}$ if $|A_i|\le a$ and $|B_i| \le b$ for each $i$.

Our central problem is to see how this bound changes with the additional condition $|A_i \cap B_j|= 1$ for $i \ne j$. Such a system is called $1$-cross intersecting. We show that the maximum size of a $1$-cross intersecting set pair system is

• at least $5^{n/2}$ for $n$ even, $a=b=n$,
• equal to $(\lfloor \frac n2 \rfloor + 1)(\lceil \frac n2\rceil + 1)$ if $a=2$, $b=n \ge 4$,
• at most $|\bigcup_{i=1}^m A_i|$,
• asymptotically $n^2$ if $\{A_i\}$ is a linear hypergraph ($|A_i\cap A_j| \le 1$ for $i \ne j$),
• asymptotically $\frac12 n^2$ if $\{A_i\}$, $\{B_i\}$ are both linear.

3:00 pm in 243 Altgeld Hall,Tuesday, October 8, 2019

#### Character stacks and shtukas in the topological setting

###### Nick Rozenblyum (U Chicago)

Abstract: I will describe a general categorical framework leading to shtukas (in the sense of Drinfeld) and excursion operators (in the sense of V. Lafforgue) on moduli spaces. In particular, I will give a concrete description of the space of functions on (derived) character varieties. I will explain how this leads to the spectral action in the context of Betti geometric Langlands and (conjecturally) to the spectral decomposition in geometric Langlands over finite fields via a categorification of Grothendieck's function-sheaf correspondence. This is joint work with Gaitsgory, Kazhdan, and Varshavsky.

Wednesday, October 9, 2019

4:00 pm in 245 Altgeld Hall,Wednesday, October 9, 2019

#### An overview of Erdős-Rothschild problems and their rainbow variants

###### Lina Li

Abstract: In 1974, Erdős and Rothchild conjectured that the complete bipartite graph has the maximum number of two-edge-colorings without monochromatic triangles over all n-vertex graphs. Since then, this new class of colored extremal problems has been extensively studied by many researchers on various discrete structures, such as graphs, hypergraphs, boolean lattices and sets. In this talk, I would like to give an overview of some past results on this topic. The second half of this talk is to investigate the rainbow variants of the Erdős-Rothschild problem. Our first main result, confirming conjectures of Benevides, Hoppen and Sampaio, and Hoppen, Lefmann, and Odermann, completes the characterization of the extremal graphs for the edge-colorings without rainbow triangles. We also studied a similar question on sum-free sets, in which we describe the extremal configurations for the colorings of integers without rainbow sums.

4:00 pm in Altgeld Hall,Wednesday, October 9, 2019

#### Disability Allyship and DRES Information

Abstract: For all faculty members and graduate students - DRES staff members will provide practical information on making classroom instruction as accessible as possible to all students and on DRES policies and procedures.

Thursday, October 10, 2019

12:00 pm in 243 Altgeld Hall,Thursday, October 10, 2019

#### You can “hear” the shape of a polygonal billiard table

Abstract: Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge-itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization.

4:00 pm in 235 Altgeld Hall,Thursday, October 10, 2019

#### Nonlocal Problems with the Fractional Laplacian and Their Applications

###### Yanzhi Zhang   [email] (Missouri University of Science and Technology)

Abstract: Recently, the fractional Laplacian has received great attention in modeling complex phenomena that involve long-range interactions. However, the nonlocality of the fractional Laplacian introduces considerable challenges in both analysis and simulations. In this talk, I will present numerical methods to discretize the fractional Laplacian as well as error estimates. Compared to other existing methods, our methods are more accurate and simpler to implement, and moreover they closely resembles the central difference scheme for the classical Laplace operator. Finally, I will show some applications of nonlocal problems involving the fractional Laplacian.

Friday, October 11, 2019

2:00 pm in 347 Altgeld Hall,Friday, October 11, 2019

###### See seminar site

Abstract: See seminar site.

3:00 pm in 341 Altgeld Hall,Friday, October 11, 2019

#### An Introduction to Groupoids in Operator Algebras

###### Vincent Villalobos (UIUC Math)

Abstract: This talk will introduce and define groupoids while motivating their importance in operator algebras. We will discuss the various structures that arise within groupoids and then explore the construction of the groupoid $C^\ast$-algebra. Finally, we will discuss the specific case of a Lie groupoid and how the added smooth structure affects the groupoid $C^\ast$-algebra.

4:00 pm in 345 Altgeld Hall,Friday, October 11, 2019

#### "Complex-like" analysis in o-minimal structures (Part 3)

###### Neer Bhardwaj (UIUC)

Abstract: Analogues of many of the basic results in complex analysis can be established over an arbitrary algebraically closed field $K$ of characteristic zero, in the context of an o-minimal expansion of a real closed field $R$, with $K=R[i]$. I will show in particular how one can define winding numbers, and how differentiability begets infinite differentiability in this setting. This expository talk follows the survey given by Lou in the last two weeks and will be based mostly on: "Expansions of algebraically closed fields in o-minimal structures" by Starchenko–Peterzil (https://link.springer.com/article/10.1007/PL00001405)

4:00 pm in Altgeld Hall 245,Friday, October 11, 2019

#### Workshop - how to write a report

###### Alfred Chong and Daniel Linders

4:00 pm in 347 Altgeld Hall,Friday, October 11, 2019

#### Kuratowski 14-Sets Theorem and Related Classifications of Topological Spaces

###### Jinghui Yang (UIUC Math)

Abstract: One very interesting theorem you may encounter when you first study topology is known as the famous Kuratowski 14-sets theorem: Given a topology space (X,T) with T is its topology, for any subset A of X, at most 14 sets can be obtained from A by taking closures and complements. This is really a shocking fact, but even more surprisingly, the “tool” to prove it is rather easy using a strong taste of algebra. Furthermore, this “tool” can be useful to do some basic classification of topology spaces; namely, you can decide the type of some topology spaces completely determined by its “K-Number”. This talk is based on the paper The Kuratowski Closure-Complement Theorem by B.J. Gardner and M. Jackson (2007), published in New Zealand Journal of Mathematics, Vol.38(2008), 9-44. Some basic concepts of topology will be reviewed.

4:00 pm in Altgeld Hall 245,Friday, October 11, 2019

#### Workshop - how to write a report

###### Alfred Chong and Daniel Linders

Monday, October 14, 2019

3:00 pm in 441 Altgeld Hall,Monday, October 14, 2019

#### Motivating Higher Toposes: Higher Bundle Theory

###### Joseph Rennie (UIUC Math)

Abstract: In this (self-contained) talk, I will begin with a quick recap of the motivation for higher bundle theory from the first talk. I will then say a few words about Toposes, and proceed to spend the majority of the talk attempting to develop a general theory of higher bundles. Along the way, we will see how the necessary properties for this development (almost) force higher topos structure. (Technical details will be sacrificed for intuitive clarity. No particular model of higher categories will be imposed.)

3:00 pm in 243 Altgeld Hall,Monday, October 14, 2019

#### Supersymmetric localization and the Witten genus

###### Dan Berwick-Evans (Illinois)

Abstract: Equivariant localization arguments generalize the Duistermaat–Heckman formula, allowing one to express an integral on a manifold in terms of integrals over fixed point sets of a torus action. Supersymmetric localization seeks to apply these formulas to path integrals in quantum field theory. In fortuitous cases, this affords a rigorous definition of the path integral. I will explain one such example in a 2-dimensional quantum field theory built on a classical theory of maps from elliptic curves to a smooth manifold. Up to a certain choice of orientation (which may be obstructed), the path integral is well-defined. The volume of the mapping space (i.e., the path integral of 1) turns out to be the Witten genus, an invariant of smooth manifolds valued in modular forms.

5:00 pm in 241 Altgeld Hall,Monday, October 14, 2019

#### The Dixmier Trace

###### Haojian Li (UIUC)

Abstract: We will construct so-called exotic traces on the space of bounded operators on a Hilbert space, the so called Dixmier traces. In the long run we are interested in geometric applications of Connes' Dixmier trace calculus.

Tuesday, October 15, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, October 15, 2019

#### Angled crested type water waves

###### Siddhant Agrawal (U Mass Amherst)

Abstract: We consider the two-dimensional water wave equation which is a model of ocean waves. The water wave equation is a free boundary problem for the Euler equation where we assume that the fluid is inviscid, incompressible, irrotational and the air density is zero. In the case of zero surface tension, we show that the singular solutions recently constructed by Wu (19) are rigid. In the case of non-zero surface tension, we construct an energy functional and prove a local wellposedness result without assuming the Taylor sign condition. This energy reduces to the energy obtained by Kinsey and Wu (18) in the zero surface tension case and allows angled crest interfaces. For non zero surface tension, the energy does not allow singularities in the interface but allows interfaces with large curvature. We show that in an appropriate regime, the zero surface tension limit of our solutions is a solution of the gravity water wave equation which includes waves with angled crests.

2:00 pm in 243 Altgeld Hall,Tuesday, October 15, 2019

#### Equiangular lines with a fixed angle

###### Zilin Jiang (MIT Math)

Abstract: An equiangular set of lines is a family of lines (through the origin) such that they are pairwise separated by the same angle. A central question in Algebraic Graph Theory is to determine the maximum cardinality of an equiangular set of lines in n-dimensional Euclidean space. In this talk, we will prove the key spectral result on the multiplicity of the second largest eigenvalue of a connected graph, and we will then connect it to the question on equiangular lines. Joint work with Jonathan Tidor, Yuan Yao, Shengtong Zhang and Yufei Zhao.

2:00 pm in 347 Altgeld Hall,Tuesday, October 15, 2019

#### On the spectral heat content for subordinate killed Brownian motions with respect to a wide class of subordinators

###### Hyunchul Park (SUNY New Paltz)

Abstract: In this talk, we study the asymptotic behavior of the spectral heat content for subordinate killed Brownian motions (SKBM) with respect to a wide class of subordinators. Previously, the spectral heat content for SKBM via stable subordinators was studied by the author and R. Song. This result gives an upper bound for the heat loss for the spectral heat content for killed Levy processes, whose asymptotic limit is not available for $\mathbb{R}^d$, $d\ge 2$, even for killed $\alpha$-stable processes when $\alpha\in [1; 2)$. This is a joint work with R. Song and is in progress.

3:00 pm in 243 Altgeld Hall,Tuesday, October 15, 2019

#### Koszul Modules and Green’s Conjecture

###### Claudiu Raicu (University of Notre Dame)

Abstract: Formulated in 1984, Green’s Conjecture predicts that one can recognize the intrinsic complexity of an algebraic curve from the syzygies of its canonical embedding. Green’s Conjecture for a general curve has been resolved in two landmark papers by Voisin in the early 00s. I will explain how the recent theory of Koszul modules provides more elementary solutions to this problem, by relating it to the study of the syzygies of some very concrete surfaces. Joint work with M. Aprodu, G. Farkas, S. Papadima, S. Sam and J. Weyman.

Thursday, October 17, 2019

11:00 am in 241 Altgeld Hall,Thursday, October 17, 2019

#### A new approach to bounds for L-functions

###### Jesse Thorner (University of Florida)

Abstract: Let $L(s)$ be the $L$-function of a cuspidal automorphic representation of $GL(n)$ with analytic conductor $C$. The Phragmen-Lindelof principle implies the convexity bound $|L(1/2)| \ll C^{1/4+\epsilon}$ for all fixed $\epsilon>0$, while the generalized Riemann hypothesis for $L(s)$ implies that $|L(1/2)|\ll C^{\epsilon}$. A major theme in modern number theory is the pursuit of subconvexity bounds of the shape $|L(1/2)| \ll C^{1/4-\delta}$ for some fixed constant $\delta>0$. I will describe how to achieve (i) an unconditional nontrivial improvement over the convexity bound for all automorphic $L$-functions (joint work with Kannan Soundararajan), and (ii) an unconditional subconvexity bound for almost all automorphic $L$-functions (joint work with Asif Zaman).

1:00 pm in 464 Loomis Laboratory ,Thursday, October 17, 2019

#### Sphere packing, modular bootstrap and extremal functionals

###### Dalimil Mazac

Abstract: I will prove a new theorem about 2D CFTs: Every unitary 2D CFT must contain a non-trivial Virasoro primary of scaling dimension at most c/8 + 1/2, where c is the central charge. At large c, this is an improvement of the Hellerman bound c/6 + O(1), and is relevant for constraining the spectrum of gravitational theories in AdS3. The proof follows from the modular bootstrap and uses analytic extremal functionals, originally developed in the context of four-point SL(2) conformal bootstrap. In the second part of the talk, I will discuss a surprising connection between modular bootstrap and the sphere-packing problem from discrete geometry. In particular, the above bound on the gap becomes a bound on the sphere-packing density. In 8 and 24 dimensions, this bound is sharp and leads to a solution of the sphere-packing problem in these dimensions, as originally proved by Viazovska et al. The talk will be based on arXiv:1905.01319.

2:00 pm in 347 Altgeld Hall,Thursday, October 17, 2019

#### Branching Processes Part 1

###### Peixue Wu (UIUC Math)

Abstract: The first part of this talk is very introductory, I will talk about the basic ideas of branching mechanism (originated from random walk) and some generalizations of the simple branching process, e.g., age-dependent processes, multi-type branching process. ​I will focus on the limit theorem of branching processes.​ ​In the second part, I will talk about the superprocess (which is measure-valued branching processes) and some recent works about it.

4:00 pm in 245 Altgeld Hall,Thursday, October 17, 2019

#### Orbit Equivalence and Entropy

###### Hanfeng Li   [email] (University at Buffalo)

Abstract: Entropy is one of the most important numerical invariants for probability-measure-preserving (pmp) actions of countable infinite groups. Orbit equivalence is a fairly weak equivalence relation between pmp actions. In general orbit equivalence may not preserve entropy. A few years ago Tim Austin showed that integrable orbit equivalence between pmp actions of finitely generated amenable groups does preserve entropy. I will introduce a notion of Shannon orbit equivalence, weaker than integrable orbit equivalence, and a property SC for pmp actions. The Shannon orbit equivalence between pmp actions of sofic groups with the property SC preserves the maximal sofic entropy. If a group G has a w-normal subgroup H such that H is amenable and neither locally finite nor virtually cyclic, then every pmp action of G has the property SC. In particular, if two Bernoulli shifts of such a sofic group are Shannon orbit equivalent, then they are conjugate. This is joint work with David Kerr.

Friday, October 18, 2019

2:00 pm in 245 Altgeld Hall,Friday, October 18, 2019

#### Careers for Math Students in the Life Sciences and Medicine

###### Howard Aizenstein, Tandy Warnow, James O'Dwyer, Olgica Milenkovic

Abstract: Are you interested in learning more about the role of mathematics in the fields of biology, biochemistry, or medicine? Come hear from a distinguished panel of applied mathematicians whose research addresses societally relevant problems in the biological sciences. Panelists: Howard Aizenstein, Charles F. Reynolds III and Ellen G. Detlefsen Endowed Chair in Geriatric Psychiatry and Professor of Bioengineering and Clinical and Translational Science at the University of Pittsburgh; Tandy Warnow, Founder Professor of Computer Science and Associate Head for Computer Science, UIUC;; James O'Dwyer, Associate Professor, Department of Plant Biology, UIUC; and Olgica Milenkovic, Professor and Donald Biggar Willett Scholar, Department of Electrical and Computer Engineering, UIUC

2:00 pm in 347 Altgeld Hall,Friday, October 18, 2019

#### Brown-Goodearl conjecture for PI weak Hopf algebras

###### James Zhang (University of Washington, Seattle)

Abstract: Brown and Goodearl conjectured that every noetherian Hopf algebra is Artin-Schelter Gorenstein. This conjecture is known to be true for many cases, in particular, for affine polynomial identity Hopf algebras. Weak Hopf algebras are an important generalization of Hopf algebras, and the category of modules over a weak Hopf algebra has a monoidal structure. Let $W$ be a weak Hopf algebra that is a finitely generated module over its affine center. We prove that $W$ has finite self-injective dimension and is a direct sum of Artin-Schelter Gorenstein algebras. Therefore Brown-Goodearl conjecture holds in this special weak Hopf setting. We will also give some motivations and consequences of Brown-Goodearl conjecture. This is joint work with Dan Rogalski and Robert Won.

4:00 pm in 141 Altgeld Hall,Friday, October 18, 2019

#### Lines in Space

###### Brian Shin (UIUC)

Abstract: Consider four lines in three-dimensional space. How many lines intersect these given lines? In this expository talk, I'd like to discuss this classical problem of enumerative geometry. Resolving this problem will give us a chance to see some interesting algebraic geometry and algebraic topology. If time permits, I'll discuss connections to motivic homotopy theory.

4:00 pm in 347 Altgeld Hall,Friday, October 18, 2019

#### Where Automata Theory Meets Metric Geometry

###### Alexi Block Gorman (UIUC Math)

Abstract: The results in this talk illustrate and expand on connections between automata theory and metric geometry. We will begin by defining automata, Buchi automata, fractals, and iterated function systems. We say that a function is regular if there is a Buchi automaton that accepts precisely the set of base n representations of points in the graph of the function. We show that a continuous regular function (with closed and bounded domain) "looks linear" almost everywhere, if you zoom in enough. As a result, we show that every differentiable regular function is a shift of linear function (or hyperplane, in higher dimensions).

Monday, October 21, 2019

3:00 pm in 441 Altgeld Hall,Monday, October 21, 2019

#### Topological Data Analysis: Theory and Applications

###### Daniel Carmody (UIUC Math)

Abstract: There are two algorithms which form the backbone of many applications of modern topological data analysis: the Mapper algorithm (Singh, Memoli, Carlsson), and persistent homology (Zomorodian, Carlsson). In this talk I'll introduce both algorithms, talk about the homotopy theory behind them, and give an application of each.

3:00 pm in 243 Altgeld Hall,Monday, October 21, 2019

#### Categories of equivariant bifurcation problems

###### Stef Klajbor-Goderich (Illinois)

Abstract: We present a framework for studying equivariant bifurcation from relative equilibria of dynamical systems with Lie group symmetries. We will explain how equivariant bifurcation problems from relative equilibria are objects of a category internal to the category of topological abelian groups, or equivalently they are part of a $2$-term chain complex of topological abelian groups. This category is topologically equivalent to a category of equivariant bifurcation problems from "honest" equilibria. Thus, generic equivariant bifurcation problems from relative equilibria are equivalent to generic equivariant bifurcation problems from equilibria. We conclude with some observations on one way an analogous approach to Hamiltonian bifurcation problems could proceed.

5:00 pm in 243 Altgeld Hall,Monday, October 21, 2019

#### Dixmier Trace

###### Haojian Li

Abstract: Diximer traces, heat kernel, and Zeta functions.

Tuesday, October 22, 2019

1:00 pm in Altgeld Hall,Tuesday, October 22, 2019

#### Orbit equivalence relations of some classes of non-locally compact Polish groups

###### Joseph Zielinski

Abstract: By results of A.S. Kechris, whenever a locally compact Polish group acts continuously on a Polish space, the orbit equivalence relation of the action is essentially countable—that is, Borel reducible to the orbit equivalence relation of an action of a countable group. It is unknown if this characterizes the locally compact Polish groups. S. Solecki, after proving an analogous characterization for smooth actions of compact Polish groups, showed this to be true in the case where the group, G, is the additive group of a separable Banach space. The characterization also holds for abelian pro-countable groups, by results of M. Malicki. We discuss recent work on this problem, including an extension of this characterization to some important classes of Polish groups. This is joint work with A.S. Kechris, M. Malicki, and A. Panagiotopoulos.

1:00 pm in 347 Altgeld Hall,Tuesday, October 22, 2019

#### The Dirichlet problem for elliptic operators having a BMO antisymmetric part

###### Linhan Li (UMN Math)

Abstract: In this talk, we are going to introduce our result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO antisymmetric part. In particular, the coefficients of the operator are not necessarily bounded. Our method relies on kernel estimates and off-diagonal estimates for the semigourp e^{-tL}, solution to the Kato problem, and various estimates for the Hardy norms of certain commutators. This is a joint work with S. Hofmann, S. Mayboroda, and J. Pipher.

2:00 pm in 243 Altgeld Hall,Tuesday, October 22, 2019

#### River landscapes and Optimal Channel Networks

###### József Balogh (Illinois Math)

Abstract: We study tree structures termed Optimal Channel Networks (OCNs) that minimize the total gravitational energy loss in the system, an exact property of steady-state landscape configurations that prove dynamically accessible and strikingly similar to natural forms. Here, we show that every OCN is a so-called natural river tree, in the sense that there exists a height function such that the flow directions are always directed along steepest descent. We also study the natural river trees in an arbitrary graph in terms of forbidden sub-structures.

The talk is based on: P. Balister, J. Balogh, E. Bertuzzo, B. Bollobas, G. Caldarelli, A. Maritan, R. Mastrandrea, R. Morris, A Rinaldo: River landscapes and Optimal Channel Networks, Proceeding of the National Academy of Sciences U.S.A., 115, 6548--6553 (2018).

This paper is based on a true collaboration among mathematicians, theoretical computer scientists and physicists.

3:00 pm in 243 Altgeld Hall,Tuesday, October 22, 2019

#### Structure of local cohomology modules associated with projective varieties

###### Wenliang Zhang (UIC)

Abstract: Let R be a polynomial ring over a field and I be an ideal of R. The local cohomology modules H^j_I(R) are rarely finitely generated as R-modules. However, they have finite length when viewed as objects in the category of D-modules in characteristic 0 (or in the category of F-modules in characteristic p). Computing the actual length (in the appropriate category) has been an open problem; it is also an open problem just to determine whether they are simple objects (in the appropriate category). In this talk, I will explain a solution to this problem when the ideal I is a homogeneous prime ideal and the projective variety Proj(R/I) has mild singularities in characteristic p. This is a joint work with Nicholas Switala.

Wednesday, October 23, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, October 23, 2019

#### Stormy actions of Bernoulli shifts

###### Joseph Zielinski

Abstract: An equivalence relation is essentially countable when it is Borel reducible to the orbit equivalence relation of some action of a countable group. In a 2005 paper, G. Hjorth introduced the notion of a stormy Polish G-space and showed that this condition is the fundamental obstruction to essential countability for orbit equivalence relations. Specifically, he proved that if a Polish group, G, acts continuously on a Polish space, X, with a Borel orbit equivalence relation then this relation fails to be essentially countable if and only there is a continuous G-equivariant embedding of a stormy action into X. In the first part of this talk we recall the basic concepts and results in Hjorth's work on stormy actions. In the second part, we apply these to the constructions considered in recent joint work with A.S. Kechris, M. Malicki, and A. Panagiotopoulos for actions of non-Archimedian groups and separable Banach spaces.

4:00 pm in 245 Altgeld Hall,Wednesday, October 23, 2019

#### Using Informal Early Feedback (IEF)

###### Lucas Anderson (Center for Innovation in Teaching & Learning )

Abstract: Most teachers get evaluated by their students at the end of the course by implementing ICES. But by then, it is too late to make changes that will make a difference for your current students. You should check in with your students before it is too late. Come to this workshop to learn how to design, implement, and interpret Informal Early Feedback (IEF) to improve the class experience for everyone.

4:00 pm in 447 Altgeld Hall,Wednesday, October 23, 2019

#### Deformations of the Frobenius

###### William Balderrama (Illinois Math)

Abstract: The deformation theory of the Frobenius homomorphism of a group scheme in positive characteristic is rich, and gives rise to various interesting algebraic structures. I'll introduce this problem, talk about the example of the multiplicative group, from which one naturally obtains Witt vectors and the notion of a delta ring, and talk about what is known about the case of elliptic curves.

Thursday, October 24, 2019

11:00 am in 241 Altgeld Hall,Thursday, October 24, 2019

#### Non-vanishing of Dirichlet L-functions

###### Rizwanur Khan (University of Mississippi)

Abstract: $L$-functions are fundamental objects in number theory. At the central point $s = 1/2$, an $L$-function $L(s)$ is expected to vanish only if there is some deep arithmetic reason for it to do so (such as in the Birch and Swinnerton-Dyer conjecture), or if its functional equation specialized to $s = 1/2$ implies that it must. Thus when the central value of an $L$-function is not a "special value", and when it does not vanish for trivial reasons, it is conjectured to be nonzero. In general it is very difficult to prove such non-vanishing conjectures. For example, nobody knows how to prove that $L(1/2, \chi)$ is nonzero for all primitive Dirichlet characters $\chi$. In such situations, analytic number theorists would like to prove 100% non-vanishing in the sense of density, but achieving any positive percentage is still valuable and can have important applications. In this talk, I will discuss work on establishing such positive proportions of non-vanishing for Dirichlet $L$-functions.

1:00 pm in 464 Loomis Laboratory,Thursday, October 24, 2019

#### Simple holographic models of black hole evaporation

###### Chris Akers (Berkeley Physics)

Abstract: Several recent papers have shown a close relationship between entanglement wedge reconstruction and the unitarity of black hole evaporation in AdS/CFT. The analysis of these papers however has a rather puzzling feature: all calculations are done using bulk dynamics which are essentially those Hawking used to predict information loss, but applying ideas from entanglement wedge reconstruction seems to suggest a Page curve which is consistent with information conservation. In this note we present a new pair of models which clarify this situation. Our first model gives a holographic illustration of unitary black hole evaporation, in which the analogue of the Hawking radiation purifies itself as expected, and this purification is reproduced by the entanglement wedge analysis. Moreover a smooth black hole interior persists until the last stages the evaporation process. Our second model gives an alternative holographic interpretation of the situation where the bulk evolution leads to information loss: unlike in the models proposed so far, this bulk information loss is correctly reproduced by the entanglement wedge analysis. In both models no bulk quantum corrections need to be considered: classical extremal surfaces are enough to do the job. We argue that our first model is a better analogy for what actually happens to evaporating black holes, but we also emphasize that any complete resolution of the information problem will require an understanding of non-perturbative bulk dynamics.

2:00 pm in 347 Altgeld Hall,Thursday, October 24, 2019

#### Branching Processes Part 2

###### Peixue Wu (UIUC Math)

Abstract: The first part of this talk is very introductory, I will talk about the basic ideas of branching mechanism (originated from random walk) and some generalizations of the simple branching process, e.g., age-dependent processes, multi-type branching process. ​I will focus on the limit theorem of branching processes.​ ​In the second part, I will talk about the superprocess (which is measure-valued branching processes) and some recent works about it.

3:00 pm in 347 Altgeld Hall,Thursday, October 24, 2019

#### Some algebraic combinatorics arising in CR Geometry

###### John P. D'Angelo   [email] (University of Illinois at Urbana-Champaign )

Abstract: We will define and discuss infinitely many triangles of integers that share many properties with Pascal’s triangle. The rows correspond to coefficients of invariant polynomials arising in CR geometry. One special case yields polynomials $f_{p,q}(x,y)$ that satisfy $f_{p,q}(x,y)$ is congruent to $x^p + y^p$ mod $(p)$ if and only if $p$ is prime or $p=1$. The talk will be accessible to beginning graduate students but will glimpse research directions.

4:00 pm in 245 Altgeld Hall,Thursday, October 24, 2019

#### Disability Allyship and DRES Information

###### Rachel Jackson Green (University of Illinois Disability Resources and Educational Services (DRES))

Abstract: This workshop is meant for all faculty and TAs - everyone who teaches courses in the Mathematics Department. Rachel Jackson Green will discuss disability allyship as it pertains to instruction. She will cover both the logistics of DRES accommodations for students as well as ways to make instruction as accessible as possible for all students, whether they are using DRES accommodations or not.

Friday, October 25, 2019

2:00 pm in 347 Altgeld Hall,Friday, October 25, 2019

#### Noncommutative algebra from a geometric point of view

###### Xingting Wang (Howard University)

Abstract: In this talk, I will discuss how to use algebro-geometric and Poisson geometric methods to study the representation theory of 3-dimensional Sklyanin algebras, which are noncommutative analogues of polynomial algebras of three variables. The fundamental tools we are employing in this work include the noncommutative projective algebraic geometry developed by Artin-Schelter-Tate-Van den Bergh in 1990s and the theory of Poisson order axiomatized by Brown and Gordon in 2002, which is based on De Concini-Kac-Priocesi’s earlier work on the applications of Poisson geometry in the representation theory of quantum groups at roots of unity. This is joint work with Milen Yakimov and Chelsea Walton.

3:00 pm in 341 Altgeld Hall,Friday, October 25, 2019

#### Continued Fractions and Ergodic theory

###### Maria Siskaki (UIUC Math)

Abstract: I will talk about how continued fractions arise. Continued fractions have had various applications in transcendental number theory and diophantine approximation . I will explain how tools from ergodic theory can be used to solve problems involving continued fractions. In particular, I will talk about the ergodic properties of the Gauss and Farey maps. The talk will be introductory.

4:00 pm in 141 Altgeld Hall,Friday, October 25, 2019

#### Geometry + Topology + Analysis + Algebra

###### Cameron Rudd (UIUC)

Abstract: Barring a last minute change of heart, this talk will be about analytic analogues of typical algebraic invariants of manifolds that have proven to be useful in understanding how geometric and topological features of aspherical Riemannian manifolds influence one another.

4:00 pm in 347 Altgeld Hall,Friday, October 25, 2019

#### Schubert Polynomials and Computational Complexity

###### Colleen Robichaux   [email] (UIUC Math)

Abstract: In this talk I will discuss recent work in Algebraic Combinatorics and how it relates to Computational Complexity. How do results in Computational Complexity influence work in Algebraic Combinatorics and vice versa? I will give introductions to both topics and discuss how they came together in joint work with Anshul Adve and Alexander Yong.

5:00 pm in 241 Altgeld Hall,Friday, October 25, 2019

#### Monotonicity of relative entropy

###### Xinan Chen (University of Illinois at Urbana-Champaign)

Abstract: Use Lieb's concavity theorem to show relative entropy is monotone decreasing under quantum channels. This is also known as data processing inequality (DPI). I will also present a short proof for subadditivity and strong subadditivity using DPI.

Monday, October 28, 2019

3:00 pm in 243 Altgeld Hall,Monday, October 28, 2019

#### Open Gromov-Witten invariants and underlying structures

###### Sara Tukachinsky (Institute for Advanced Study)

Abstract: For $X$ a symplectic manifold and $L$ a Lagrangian submanifold, genus zero open Gromov-Witten (OGW) invariants count configurations of pseudoholomorphic disks in X with boundary conditions in L and various constraints at boundary and interior marked points. In a joint work with Jake Solomon from 2016, we define OGW invariants using bounding chains, a concept that comes from Floer theory. In a recent work, also joint with Solomon, we find that the generating function of OGW satisfies a system of PDE called open WDVV equation. This PDE translates to an associativity relation for a quantum product we define on the relative cohomology $H^*(X,L)$. For the projective space, open WDVV gives rise to recursions that, together with other properties, allow the computation of all OGW invariants.

3:00 pm in 441 Altgeld Hall,Monday, October 28, 2019

#### Global homotopy groups and global functors

###### Heyi Zhu (UIUC Math)

Abstract: The 0th equivariant homotopy group of an orthogonal $G$-spectrum defines a $G$-Mackey functor with restriction and transfer functors out of the the orbit category of $G$ and when $G$ is finite, the interactions between restrictions and transfers is given by a double coset formula. In the global setting, we define an orthogonal spectrum as a functor out of inner product spaces and will see that its 0th global homotopy group defines a "global functor" out of the global Burnside category so that restriction generalizes naturally for arbitrary continuous maps of groups and transfer along inclusion of closed subgroups. In this sense, the global functors generalize $G$-Mackey functors by allowing $G$ to vary. This talk follows the treatment in Schwede's book Global homotopy theory.

5:00 pm in 241 Altgeld Hall ,Monday, October 28, 2019

#### The Dixmier trace

###### Haojian Li (UIUC)

Abstract: The interplay between heat kernel functional, zeta function and the Dixmier trace. If time permits, I would also introduce pseudo differential operator.

Tuesday, October 29, 2019

11:00 am in 347 Altgeld Hall,Tuesday, October 29, 2019

#### Topological Quillen localization of structured ring spectra

###### Yu Zhang (Ohio State University)

Abstract: Homotopy groups and stable homotopy groups of spaces are main invariants in algebraic topology. Homotopy groups are very powerful but difficult to compute in practice. Stable homotopy groups, on the other hand, are easier to work with, at the expense of losing unstable information. Structured ring spectra are spectra with certain algebraic structure encoded by the action of an operad O. For such O-algebras, the analog of stable homotopy groups is played by Topological Quillen (TQ) homology groups. In this talk, we will discuss the following question: What information can be seen by TQ-homology? In particular, we will discuss TQ-localization of O-algebras and show the TQ-Whitehead theorem for homotopy pro-nilpotent O-algebras.

1:00 pm in 347 Altgeld Hall,Tuesday, October 29, 2019

#### The Abnormally Normal Behavior of the Nonlinear Schroedinger Equation

###### Katelyn Leisman (Illinois Math)

Abstract: The Nonlinear Schroedinger Equation (NLS) is an important partial differential equation that models many different physical applications, including super-fast lasers, Bose-Einstein condensates, and light traveling in optical fibers and wave guides. The goal in studying this equation is to know how its solutions (and thus the physical systems they model) behave over time. One way to do this for linear ("normal") equations is by finding a relationship between the wavelength and the frequency, called the dispersion relation. Unfortunately, the traditional dispersion relation approach does not work for nonlinear waves (like the NLS). However, I've found that some numerical solutions of the NLS have an effective dispersion relation. In this talk, I'll discuss this important equation and this apparent abnormally "normal" behavior. This talk will be accessible to a general audience.

2:00 pm in 243 Altgeld Hall,Tuesday, October 29, 2019

#### A bandwidth theorem for locally dense graphs

###### Andrew Treglown (University of Birmingham Math)

Abstract: A fundamental topic in extremal graph theory is to find minimum degree conditions that force a spanning substructure in a graph. One of the most general results in this direction is the so-called Bandwidth Theorem of Boettcher, Schacht and Taraz. This result gives a minimum degree condition which forces a graph G to contain every spanning subgraph of bounded chromatic number, bounded degree and sublinear bandwidth. In this talk I will describe a version of the Bandwidth Theorem where now one substantially lowers the degree condition at the expense of ensuring the host graph G is "locally dense". This is joint work with Katherine Staden.

Wednesday, October 30, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, October 30, 2019

#### On uniform ergodic decomposition

###### Ruiyuan (Ronnie) Chen (UIUC Math)

Abstract: We will prove Farrell–Varadarajan's uniform ergodic decomposition theorem for countable Borel equivalence relations. The proof we give is based on using the Becker–Kechris comparability lemma to construct uniform conditional probability functions for Borel sets.

4:00 pm in 447 Altgeld,Wednesday, October 30, 2019

#### A Case for Étale Cohomology

###### Justin Kelm (Illinois Math)

Abstract: This talk will serve as an introduction to étale cohomology. In particular, I will focus on the shortcomings of Zariski cohomology, why étale maps are a good algebro-geometric substitute for the notion of a "covering map" or "local" diffeomorphism, the technology that étale cohomlogy is built out of, and a basic computation or two that should illuminate why étale cohomology should give rise to a useful Weil cohomology theory.

Thursday, October 31, 2019

11:00 am in 241 Altgeld Hall,Thursday, October 31, 2019

#### Eisenstein ideal with squarefree level

###### Carl Wang-Erickson (University of Pittsburgh)

Abstract: In his landmark paper "Modular forms and the Eisenstein ideal," Mazur studied congruences modulo a prime p between the Hecke eigenvalues of an Eisenstein series and the Hecke eigenvalues of cusp forms, assuming these modular forms have weight 2 and prime level N. He asked about generalizations to squarefree levels N. I will present some work on such generalizations, which is joint with Preston Wake and Catherine Hsu.

12:00 pm in 243 Altgeld Hall,Thursday, October 31, 2019

#### Monopole Floer homology and spectral geometry of hyperbolic three-manifolds

###### Francesco Lin (Columbia University)

Abstract: By studying the Seiberg-Witten equations, Kronheimer and Mrowka defined a package of invariants of three-manifolds called monopole Floer homology. In this talk, we discuss some interactions between this topological invariant and the spectral geometry of the Laplacian on the underlying Riemannian manifold, with the goal of understanding concrete examples of hyperbolic manifolds. This is joint work with Mike Lipnowski.

1:00 pm in 464 Loomis Laboratory,Thursday, October 31, 2019

#### Multitrace excited states and perturbative entropy divergences

###### Antony Speranza (Perimeter Institute)

Abstract: Perturbative calculations of entanglement entropy have found a number of recent applications in understanding field theory, holography, and quantum gravity. In this talk, I will discuss a class of excited states of a CFT formed from a Euclidean path integral with nonlocal multitrace insertions, with an eye toward understanding their entanglement structure holographically. Such states are argued to have good semiclassical holographic duals, but can possess nontrivial bulk entanglement structure at order N^0. A surprising feature of these states is that divergences occur quite generically in the perturbative expansion of their entanglement entropy. These divergences signal a nonanalyticity in the expansion, and must be resummed in computing the full expression for the entropy. This resummation is challenging in general, but I will describe some simplified examples in which it may be tractable. In the process, I will also give some general techniques for diagnosing when such nonanalyticities occur, and point to some indications that they may in general be calculable. This talk is based on arXiv:1904.01584 and ongoing work.

2:00 pm in 243 Altgeld Hall,Thursday, October 31, 2019

#### Matrix Convex Sets, Tensor Products, and Noncommutative Choquet Boundaries

###### Roy Araiza (Purdue )

Abstract: I will discuss tensor products in the category of matrix convex sets and discuss how we may relate the study of their Choquet theory to questions about tensor product nuclearity. Based on joint work with Adam Dor-On and Thomas Sinclair.

3:00 pm in 347 Altgeld Hall,Thursday, October 31, 2019

#### The combinatorics of spherical Schubert varieties

###### Reuven Hodges   [email] (University of Illinois at Urbana-Champaign )

Abstract: Over the last several years, in joint work with V. Lakshmibai and M. B. Can, I have been studying Levi subgroup actions on Schubert varieties. I plan to discuss some of the combinatorial questions that have arisen from this work. I will introduce skew Schur functions and their decomposition in the basis of Schur functions, as well as when these decompositions are multiplicity free. Then we will discuss several problems involving these multiplicities. To conclude, I will illustrate how the answer to these multiplicity problems allows us to classify all Schubert varieties in the Grassmannian and Levi subgroups such that the Levi acts spherically on the Schubert variety.

Friday, November 1, 2019

2:00 pm in 347 Altgeld Hall,Friday, November 1, 2019

#### Categorification and quantum symmetry

###### Colleen Delaney (Indiana University)

Abstract: One variation on the theme of quantum symmetry" is a categorical group action on a unitary modular tensor category, which can be interpreted physically as a global symmetry of a 2-dimensional topological quantum phase of matter. Much of our understanding of tensor category theory and hence topological phases comes from categorification: from generalizing theorems we have about rings to theorems about categories. For example, categorifying an easy theorem in commutative ring theory, the work of Etingof, Nikshych, and Ostrik established an equivalence between categorical G-actions on modular tensor categories (MTCs), and so-called G-crossed braided extensions of MTCs. Physicists Barkeshli, Bonderson, Cheng, and Wang then recognized that this correspondence can be understood as a tensor-categorical formulation of gauge coupling, wherein G-crossed braided extensions of MTCs give an algebraic theory of symmetry-enriched topological (SET) phases of matter. While the abstract theory of Etingof, Nikshych, and Ostrik is well understood, even constructing the de-categorified part of G-crossed braided extensions of MTCs, namely their fusion rings, is challenging problem in general. We will give a two-part talk, starting with an introduction to the algebraic theory of SET phases described above. In the second part of the talk we describe a topological phase-inspired approach to constructing the fusion rings of certain G-crossed extensions called permutation extensions and share work in progress with E. Samperton in constructing their categorifications.

3:00 pm in 341 Altgeld Hall,Friday, November 1, 2019

#### Connections of Fixed-Point Theorems with Complexity

###### Basilis Livanos (UIUC CS)

Abstract: In this talk, we study how fixed-point theorems arise in the field of complexity theory and their deep connections with complexity classes like TFNP which contain problems who are guaranteed to have a solution. In the process, we provide an introduction to complexity theory and also a generalization of Bessaga's and Meyers's converse theorems to Banach's fixed-point theorem. The talk will be introductory and no prior knowledge of complexity theory will be needed.

4:00 pm in 345 Altgeld Hall,Friday, November 1, 2019

#### O-minimal complex analysis (Part 4)

###### Elliot Kaplan (UIUC)

Abstract: I will continue discussing Peterzil and Starchenko's treatment of definable functions on the algebraic closure of an o-minimal field.

4:00 pm in 141 Altgeld Hall,Friday, November 1, 2019

#### Flexibility vs Rigidity in hyperbolic geometry

###### Xiaolong Han (UIUC)

Abstract: "Most" closed surfaces have a hyperbolic structure. We can ask similar questions in higher dimensions. In this talk, I will talk about some interesting phenomena and duality by looking at examples and theorems in hyperbolic geometry. I will also talk about rigidity, like how having isomorphic first fundamental group implies existence of diffeomorphism / isometry. The talk will mostly use intuition and requires no prior knowledge of hyperbolic geometry.

4:00 pm in 347 Altgeld Hall,Friday, November 1, 2019

#### Games That Take Forever (Literally)

###### Jenna Zomback   [email] (UIUC Math)

Abstract: Do you have a winning strategy for playing tic tac toe? What about chess? In a two player game (with no ties), we say that Player 1 has a winning strategy if she can always make sure that she wins the game, regardless of what Player 2 does. In this talk, we will make this definition a bit more formal, and we'll prove that in any game that ends after finitely many steps, one of the players has a winning strategy. We'll also discuss infinite games (that end after infinitely many steps), and what it means to have a winning strategy in these games. If time allows, we'll prove that in special types of infinite games, one of the players has a winning strategy.

5:00 pm in 241 Altgeld Hall,Friday, November 1, 2019

#### Stochastic Differential Equations and Unitaries

###### Marius Junge (University of Illinois at Urbana-Champaign)

Abstract: TBD

Monday, November 4, 2019

3:00 pm in 441 Altgeld Hall,Monday, November 4, 2019

#### Atiyah-Segal completion theorem

###### Tsutomu Okano (UIUC Math)

Abstract: In this talk I will walk through Atiyah and Segal's "Equivariant K theory and completion". The main result is a nice proof of the so-called Atiyah-Segal completion theorem, which relates the equivariant K theory of a G-space with the K theory of its homotopy orbit. Towards the end, I will also discuss the algebraic analogue of this result.

3:00 pm in 243 Altgeld Hall,Monday, November 4, 2019

#### Toric degeneration and symplectic rigidity

###### Susan Tolman (Illinois)

Abstract: This talk is based on joint work with Milena Pabiniak. We say that a family of symplectic manifolds satisfies symplectic rigidity if they are classified up to symplectomorphism by their cohomology ring and the cohomology class of the symplectic form. We show how toric degeneration can be used to construct new symplectomorphisms between certain smooth toric manifolds, and then use this to show that symplectic rigidity holds for a large family of Bott manifolds.

5:00 pm in 241 Altgeld Hall,Monday, November 4, 2019

#### The Dixmier trace

###### Haojian Li (University of Illinois at Urbana-Champaign)

Abstract: I would introduce pseudo-differential operators and noncommutative residues. The calculus of pseudo-differential operators is the prototype of Alain Connes' quantum calculus in noncommutative geometry. Mariusz Wodzicki defined the non-commutative residue on the classical compactly supported pseudo differential operators by integrating the principal symbol. Connes gave the first result to identify Wodzicki residue with a singular trace, i.e., so-called Dixmier trace. This result is considered as the foundation of the noncommutative geometry.

Tuesday, November 5, 2019

11:00 am in 347 Altgeld Hall,Tuesday, November 5, 2019

#### Loop space constructions of elliptic cohomology

###### Matthew Spong

Abstract: Elliptic cohomology is a type of generalised cohomology theory related to elliptic curves which was introduced in the late 1980s. An important motivation for its introduction, which came from physics, was to help understand index theory for families of differential operators over free loop spaces. Yet for a long time, the only known constructions of elliptic cohomology were purely algebraic, and the precise connection to free loop spaces remained obscure. In this talk, I will summarise two constructions of complex analytic, equivariant elliptic cohomology: one from the K-theory of free loop spaces, and one from the ordinary cohomology of double free loop spaces. If time permits, I will also describe the construction of a Chern character-type map from the former to the latter.

1:00 pm in 347 Altgeld Hall,Tuesday, November 5, 2019

#### A bilinear proof of decoupling for the quartic moment curve

###### Zane Li (Indiana Math)

Abstract: Using a bilinear method inspired from Wooley's nested efficient congruencing method, we prove a sharp $l^2 L^{20}$ decoupling inequality for the moment curve in $\mathbb{R}^4$. This is joint work with Shaoming Guo, Po-Lam Yung, and Pavel Zorin-Kranich.

1:00 pm in 345 Altgeld Hall,Tuesday, November 5, 2019

#### Deciding the theories of expansions of the real ordered group via Ostrowski numeration

###### Christian Schulz (UIUC Math)

Abstract: For which irrational numbers $\alpha$ does the theory of $(\mathbb{R}, <, +, \mathbb{Z}, \alpha \mathbb{Z})$ have a decision algorithm? Previously, this was known for quadratic $\alpha$ thanks to work by Hieronymi. In this talk, we present the progress so far on generalizing this result to non-quadratic $\alpha$. We also discuss applications of this work to the study of combinatorics on words using automated theorem-proving software. We end with a discussion of potential future work and goals.

2:00 pm in 243 Altgeld Hall,Tuesday, November 5, 2019

#### Super-pancyclic hypergraphs and bipartite graphs

###### Dara Zirlin (Illinois Math)

Abstract: We find Dirac-type sharp sufficient conditions for a hypergraph $H$ with few edges to have a hamiltonian Berge cycle. Furthermore, these conditions yield that $H$ is super-pancyclic, i.e., for each $A \subseteq V(H)$ with $|A| \ge 3$, $H$ contains a Berge cycle with vertex set $A$. To do this, we exploit the language of bipartite graphs. In particular, we extend some results of Jackson on the existence of long cycles in bipartite graphs where the vertices in one part have high degrees, and prove his conjecture from 1981 on the topic.

This is joint work with Alexandr Kostochka and Ruth Luo.

2:00 pm in 347 Altgeld Hall,Tuesday, November 5, 2019

#### A Probabilistic Intuitive Boundary Construction

###### Peter Loeb (Illinois Math)

Abstract: Robert Martin’s 1941 generalization of the boundary of the unit disk is now a fundamental tool in potential theory and probability theory. After an introduction for the non-specialist, I will present an alternative approach to Martin’s construction and integral representation. The approach looks inside a domain using Brownian paths starting at nonstandard points that merge to form boundary points.

3:00 pm in 243 Altgeld Hall,Tuesday, November 5, 2019

#### Rational points and derived equivalence

###### Ben Antieau (UIC)

Abstract: Suppose that X and Y are smooth projective varieties over a field k and suppose that X and Y have equivalent derived categories of sheaves. If X has a rational point, does Y have a rational point? This question was asked 10 years ago by Esnault. I will report on joint work with Addington, Frei, and Honigs which shows that, in general, the answer is ‘no’, in contrast to what happens for curves (Antieau—Krashen—Ward) or in dimension at most 3 over finite fields (Honigs).

Wednesday, November 6, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, November 6, 2019

#### A pointwise ergodic theorem for hyperfinite equivalence relations

###### Jenna Zomback (UIUC Math)

Abstract: An equivalence relation on a standard Borel space is called hyperfinite if it is a countable increasing union of Borel equivalence relations whose classes are finite. We will prove the natural pointwise ergodic theorem for probability measure preserving hyperfinite equivalence relations, as recorded by Miller and Tserunyan in [MTs17]. If time permits, Anush Tserunyan will continue with a prelude to the next talk on proving the existence of a uniformly ergodic hyperfinite subequivalence relation of a given pmp countable Borel equivalence relation.

4:00 pm in 245 Altgeld Hall,Wednesday, November 6, 2019

#### Framed cobordism and the $J$-homomorphism

###### Ningchuan Zhang

Abstract: One of the main goals in homotopy theory is to compute the stable homotopy groups of spheres. Geometrically, these groups classify cobordism classes of framed submanifolds of spheres. This is a generalization of degrees of maps between oriented manifolds. From this perspective, I will explain in this talk how to compute the first stable homotopy group of spheres. This computation leads to the $J$-homomorphism, whose image is the most well-understood part in the stable homotopy groups of spheres. I will define the $J$-homomorphism using framed cobordism. If time allows, I will also give some computational results of its image, which are related to the Bernoulli numbers.

Thursday, November 7, 2019

11:00 am in 241 Altgeld Hall,Thursday, November 7, 2019

#### An even parity instance of the Goldfeld conjecture

###### Ashay Burungale (Caltech)

Abstract: We show that the even parity case of the Goldfeld conjecture holds for the congruent number elliptic curve. We plan to outline setup and strategy (joint with Ye Tian).

12:00 pm in 243 Altgeld Hall,Thursday, November 7, 2019

#### Trees, dendrites, and the Cannon-Thurston map

###### Elizabeth Field (Illinois Math)

Abstract: When $1 \to H \to G \to Q \to 1$ is a short exact sequence of three word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from $H$ to $G$ extends continuously to a map between the Gromov boundaries of $H$ and $G$. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point $z$ in the Gromov boundary of $Q$ an ending lamination'' on $H$ which consists of pairs of distinct points in the boundary of $H$. We prove that for each such $z$, the quotient of the Gromov boundary of $H$ by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where $H$ is a free group and $Q$ is a convex cocompact purely atoroidal subgroup of Out($F_n$), one can identify the resultant quotient space with a certain $\mathbb R$-tree in the boundary of Culler-Vogtmann's Outer space.

1:00 pm in 347 Altgeld Hall,Thursday, November 7, 2019

#### Theoretical and Empirical Advances in Large-Scale Species Tree Estimation

###### Tandy Warnow (Computer Science, University of Illinois)

Abstract: The estimation of the "Tree of Life" -- a phylogeny encompassing all life on earth--is one of the big Scientific Grand Challenges. Maximum likelihood (ML) is a standard approach for phylogeny estimation, but estimating ML trees for large heterogeneous datasets is challenging for two reasons: (1) ML tree estimation is NP-hard (and the best current heuristics can use hundreds of CPU years on relatively small datasets, just to find local optima), and (2) the statistical models used in ML tree estimation methods are much too simple, failing to acknowledge heterogeneity across genomes or across the Tree of Life. These two "big data" issues -- dataset size and heterogeneity -- impact the accuracy of phylogenetic methods and have consequences for downstream analyses. In this talk, I will describe a new graph-theoretic "divide-and-conquer" approach to phylogeny estimation that addresses both types of heterogeneity. Our protocol operates as follows: (1) we divide the set of species into disjoint subsets, (2) we construct trees on the subsets (using appropriate statistical methods), and (3) we combine the trees together using auxiliary information, such as a matrix of pairwise distances. I will present three such strategies (all published in the last year) that operate in this fashion, and that improve the theoretical and empirical performance of phylogeny estimation methods. One of the main applications of this work is species tree estimation from multi-locus data sets when gene trees can differ from the species tree due to incomplete lineage sorting. This talk is largely based on joint work with my PhD students, Erin Molloy and Vladimir Smirnov (Illinois).

2:00 pm in 243 Altgeld Hall,Thursday, November 7, 2019

#### Noncommutative strong maximals and almost uniform convergence in several directions

Abstract: We revisit Miguel de Guzmán's proof of the strong maximal theorem and we give an argument than can be generalized to noncommutative probability spaces. As is usual in this context, one difficulty that we must face is the definition of the operator itself: in principle, one cannot make sense of the supremum of a sequence of positive operators, something that can be done pointwise in the case of functions. We shall also discuss applications to almost everywhere convergence of sequences of averaging operators in von Neumann algebras.

4:00 pm in 245 Altgeld Hall,Thursday, November 7, 2019

#### A survey of Sperner theory

###### Richard Stanley   [email] (MIT and University of Miami)

Abstract: Let $X$ be a collection of subsets of an $n$-element set $S$ such that no element of $X$ is a subset of another. In 1927 Emanuel Sperner showed that the number of elements of $X$ is maximized by taking $X$ to consist of all subsets of $S$ with $\lfloor n/2\rfloor$ elements. This result started the subject of \emph{Sperner theory}, which is concerned with the largest subset $A$ of a finite partially ordered set $P$ that forms an \emph{antichain}, that is, no two elements of $A$ are comparable in $P$. We will give a survey of Sperner theory, focusing on some connections with linear algebra and algebraic geometry.

Friday, November 8, 2019

2:00 pm in 347 Altgeld Hall,Friday, November 8, 2019

#### Cohomology for Hopf algebras and dualities

###### Cris Negron (University of North Carolina)

Abstract: In studies of finite-dimensional Hopf algebras one makes consistent use of a number of standard operations. The most fundamental of these operations include linear duality, (cocycle) deformation, and twisting by so-called Drinfeld twists. Many numerical invariants of Hopf algebras are known to be stable under these operations. However, one can see from examples, that the cohomology ring H*(A,k) for a finite-dimensional Hopf algebra A with trivial coefficients is not preserved under deformation or duality of A. In joint work with J. Plavnik we conjecture that, although the cohomology ring itself may vary, the Krull dimension of cohomology should be invariant under a general class of duality operations" which includes linear duality, deformation, and Drinfeld twisting. I this talk I will give the necessary definitions and examples, discuss the aforementioned conjecture, and provide some positive results obtained jointly with J. Plavnik.

3:00 pm in Illini Hall 1,Friday, November 8, 2019

#### On two central binomial series related to $\zeta(4)$

###### Vivek Kaushik (UIUC)

Abstract: In this expository talk, we prove two related central binomial series identities: $\sum_{n \geq 0} \frac{{2n}\choose{n}}{2^{4n}(2n+1)^3}=\frac{7 \pi^3}{216}$ and $\sum_{n \in \mathbb{N}} \frac{1}{{n^4}{{2n}\choose{n}}}=\frac{17 \pi^4}{3240}.$ These series resist all standard approaches used to evaluate other well-known series such as the Dirichlet $L$ series. Our method to prove these central binomial series identities in question will be to evaluate two log-sine integrals that are equal to the series representations. The evaluation of these log-sine integrals will lead to computing closed forms of polylogarithms evaluated at certain complex exponentials. After proving our main identities, we discuss some polylogarithmic integrals that can be readily evaluated using the knowledge of these central binomial series.

4:00 pm in 345 Altgeld Hall,Friday, November 8, 2019

#### O-minimal complex analysis (Part 5)

###### Elliot Kaplan (UIUC)

Abstract: I will continue discussing Peterzil and Starchenko's treatment of definable functions on the algebraic closure of an o-minimal field.

4:00 pm in 141 Altgeld Hall,Friday, November 8, 2019

#### Finite Element Exterior Calculus

###### Nikolas Wojtalewicz (UIUC)

Abstract: In this talk, we will begin by discussing a basic example of a finite element method. We will state the basic formulation of this method, and then briefly discuss some of its limitations. We will follow up by talking about Hilbert complexes (such as the De Rahm complex), discretizing such complexes, and then about Finite Element Exterior Calculus. If time permits, we will show some examples where FEEC has been particularly successful.

4:00 pm in 347 Altgeld Hall,Friday, November 8, 2019

#### EG Tableaux and Complexity

###### Anna Chlopecki & Jackie Oh (UIUC Math/UIUC Computer Science)

Abstract: The purpose of this talk is to examine a counting problem in algebraic combinatorics. We will be discussing connections between reduced words, Young tableaux, and the Lascoux-M.-P. Schützenberger transition algorithm in hopes to provide an intuition for proving that counting the number of Edelman Greene tableaux for a given permutation w and partition λ is in #P.

5:00 pm in 241 Altgeld Hall,Friday, November 8, 2019

#### Stochastic Differential Equations and Unitaries

###### Marius Junge (University of Illinois at Urbana-Champaign)

Abstract: Part II.

Saturday, November 9, 2019

9:00 am in 245 Altgeld Hall,Saturday, November 9, 2019

#### Quasy-Con

###### Various regional speakers (various)

Abstract: "Quasy-Con" is an informal Quantum Symmetries Conference in the U.S. Midwest, which will be held on November 9-10, 2019. See site for more details: https://faculty.math.illinois.edu/~notlaw/QuaSy-Con2019.html.

Monday, November 11, 2019

3:00 pm in 441 Altgeld Hall,Monday, November 11, 2019

#### Introduction to Picard groups

###### Venkata Sai Narayana Bavisetty (UIUC Math)

Abstract: Picard groups have been classically studied in commutative algebra and algebraic geometry. These can be suitably generalised to define picard groups of $\mathbb{E}_{\infty}$ ring spectra. In this situation a natural question to ask is "How is the picard group of $R$ related to the picard group of $R_*$?". In this expository talk, I will explain some methods which have been used to answer the above question.

5:00 pm in 241 Altgeld Hall,Monday, November 11, 2019

#### CLSI of graphs

###### Nicholas LaRacuente (University of Illinois at Urbana-Champaign)

Abstract: Quantum Markov semigroups (QMSs) model continuous time-evolution of quantum states exposed to environmental noise. A key aspect of a selfadjoint QMS is how quickly an arbitrary state or density decays toward an invariant fixed point algebra. A modified logarithmic Sobolev inequality (MLSI) quantifies exponential decay of relative entropy with respect to the fixed point algebra. A complete logarithmic Sobolev inequality (CLSI) does so in the presence of an arbitrary auxiliary system that is untouched by decay. I will describe our recent proof that error models described by finite graphs have CLSI. This follows joint work with Marius Junge and Haojian Li.

Tuesday, November 12, 2019

11:00 am in 347 Altgeld Hall,Tuesday, November 12, 2019

#### Equivariant symmetric monoidal categories and K-theory

###### Peter Bonventre (University of Kentucky)

Abstract: Symmetric monoidal categories have (at least) two allures to homotopy theorists: they describe the algebraic structure appearing in many categories of interest, and they effectively model all connective spectra. Equivariantly, both of these tasks become more interesting, as the category of (connective) genuine equivariant spectra is significantly more subtle. In this talk, I introduce a new model for equivariant symmetric monoidal categories which both describes the algebraic structure in equivariant categories and has a K-theory functor to genuine G-spectra. I will give several examples and applications, including comparisons to many previously proposed models of genuine symmetric monoidal categories and equivariant algebra.

11:00 am in 347 Altgeld Hall,Tuesday, November 12, 2019

#### Equivariant symmetric monoidal categories and K-theory

###### Peter Bonventre (University of Kentucky)

Abstract: Symmetric monoidal categories have (at least) two allures to homotopy theorists: they describe the algebraic structure appearing in many categories of interest, and they effectively model all connective spectra. Equivariantly, both of these tasks become more interesting, as the category of (connective) genuine equivariant spectra is significantly more subtle. In this talk, I introduce a new model for equivariant symmetric monoidal categories which both describes the algebraic structure in equivariant categories and has a K-theory functor to genuine G-spectra. I will give several examples and applications, including comparisons to many previously proposed models of genuine symmetric monoidal categories and equivariant algebra.

1:00 pm in 345 Altgeld Hall,Tuesday, November 12, 2019

#### Model theory of $\mathbb{R}$-trees and of ultrametric spaces

###### Ward Henson (UIUC Math)

Abstract: First, we consider the class of metric spaces $(M,d)$ that are $\mathbb{R}$-trees with a convex metric. To treat this class using continuous first order logic, we fix a base point $p$ in $M$ and require that $M$ have radius at most $r$ with respect to $p (r>0)$. The class of these structures $(M,d,p)$ is axiomatizable. Moreover, the theory of this class has a model companion $T$, whose models we describe precisely. This theory is a well behaved continuous theory. For example, $T$ has QE and is complete; it is stable (but not superstable) and has the maximum possible number of models in each infinite cardinal.
Second, given a model $M = (M,d,p)$ of $T$, we consider the closed subset $E_r(M) := \{x \in M | d(p,x)=r\}$. This is a definable set for T, and the entire structure $M$ can be reconstructed from $(E_r(M),d)$. The metric $d$ on $E_r(M)$ is an ultrametric; further, at every $x \in E_r(M)$, the set of distances $\{d(x,y) | y \in E_r(M)\}$ is dense in the interval $[0,2r]$. These properties are easily seen to be axiomatizable in continuous logic, and we let $T^*$ denote the resulting theory. We show that $T^*$ has QE, so it is complete; further, $T$ and $T^*$ are bi-interpretable.
This is joint work with Sylvia Carlisle.

1:00 pm in 347 Altgeld Hall,Tuesday, November 12, 2019