Mathematics

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.)

Organizational Meeting

Sungwoo Nam (UIUC Math)

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.

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.

Organizational Meeting

Derek Kielty (Illinois Math)

Organizational Meeting

Nachiketa Adhikari (UIUC)

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!

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

To Be Announced

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.

Quantum Null Energy Conjecture

Tom Faukner (UIUC physics)

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

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).

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].

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.

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.

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.

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/

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 $Z$.

Canceled

Rescheduled

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.

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.

A Heat Trace Anomaly on Polygons

Hadrian Quan (Illinois Math)

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.

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.

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.

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.

A brief introduction to differential and Riemannian geometry

Adam Dor-On (UIUC)

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.

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.

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.

Non-reduced Parabolic Group Schemes, II

William Haboush (UIUC Math)

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.

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.

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.

"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].

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.

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.

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.

"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.

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.

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.

Introduction to differential and Riemannian geometry part II

Adam Dor-On (UIUC)

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.

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.

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.

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.

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.

To Be Announced

Chen Chen (University of Chicago, Geophysical Sciences)

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.

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.

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.

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.

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.

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.

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.

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.

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.

To Be Announced

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.

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.

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.

Introduction to differential and Riemannian geometry part III

Adam Dor-On (UIUC)

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.

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.

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.

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.

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."

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?

To Be Announced

Elisabeth Moyer (University of Chicago, Geophysical Sciences)

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

Cancelled

Curvature in Riemannian geometry

Adam Dor-On (UIUC)

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.

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.

n-dependent groups and fields

Nadja Hempel (UCLA)

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-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.

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.

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.

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.

"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].

To Be Announced

Sooin Yun (University of Illinois, Statistics)

My Grad School Journey

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.

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.

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.

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.

"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.

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.

Curvatures of Riemannian Lie groups

Adam Dor-On (UIUC)

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.

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.

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.

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.

Altgeld-Illini Renovation/Building Project Feedback Session

Abstract: CANCELED

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.

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.

To Be Announced

Ben Vega Westhoff (University of Illinois, Atmospheric Sciences)

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.

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.

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.

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.

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.

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.

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.

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.

Organizational meeting

Abstract: Visiting day for admitted PhD students.

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.

Riemannian geometry and Clifford algebras

Nijholt, Eeltje Cornelis (UIUC)

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.

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.

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.

To Be Announced

TBA

Altgeld-Illini Renovation/Building Project Feedback Session

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

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.

To Be Announced

Trevor Harris (University of Illinois, Statistics)

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.

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.

Please come and support our amazing undergraduate scholars!

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
Higher Shuffles Operads
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

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.

SL(3,C), SU(2,1) and the Whitehead link complement

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.

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.

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.

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").

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.

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.

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.

Riemannian geometry and Clifford algebras II

Nijholt, Eeltje Cornelis (UIUC)

To Be Announced

Cancelled

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.

No Seminar This Week

TBA

Altgeld-Illini Renovation/Building Project Feedback Session

Abstract: This feedback session will focus on classrooms and technology

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.

To Be Announced

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

"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 $\mathrm{Aut}(\mu)$.

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.

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.

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)

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.

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.

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.

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.

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).

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.

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.

Generalizing Koopman Theory to Allow for Inputs and Control

Kim, Hee Yeon (University of Illinois )

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.

Operadic Monoidal Structures

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.

Weil-Petersson analogs for graphs

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.

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.

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

Alfredo Wetzel (Wisconsin-Madison)

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.

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.

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).

Department Awards Ceremony

To Be Announced

Alfredo Wetzel (University of Wisconsin-Madison, Mathematics)

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.

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.

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.

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 $d>8$.

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.

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.

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!

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!

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

Intersection Theory III - Chern classes of vector bundles

Nachiketa Adhikari (Illinois Math)

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.

To Be Announced

Danielle Sass (University of Illinois, Statistics)

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.

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.

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.

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."

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.

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.

To Be Announced

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.

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.)

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.

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.

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.

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.

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.

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.

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.

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 ${\mathbb C}^n$.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Cancelled

(UIUC Math)

Introduction to Generating Functions

Longzheng Chen (UIUC Math)

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.

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.

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.

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.

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.

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.

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.

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?

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.

To Be Announced

Mary Silber (University of Chicago, Statistics)

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.

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

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.

Classification of irreversible and reversible operator algebras

Adam Dor-On (UIUC Math)

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.

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.