Department of

# Mathematics

Seminar Calendar
for events the week of Wednesday, September 18, 2019.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, September 16, 2019

3:00 pm in 441 Altgeld Hall,Monday, September 16, 2019

#### Modeling higher algebra with product-and-loop theories

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

Abstract: In this talk, I will introduce the extra homotopical properties of a (suitably infinitary) algebraic theory that make it suitable for modeling spectral, or otherwise higher, algebra, rather than merely derived forms of ordinary algebra. To illustrate the utility of this viewpoint, I will indicate some of the computational tools that can be constructed and understood from this perspective. Time permitting, I will discuss some applications to chromatic homotopy theory.

3:00 pm in 243 Altgeld Hall,Monday, September 16, 2019

#### Packing Lagrangian Tori

###### Ely Kerman (Illinois)

Abstract: While Lagrangian tori bound no volume in dimensions greater than two, in some ways, they behave as if they do. For example, it takes a nontrivial amount of energy to displace one. In this spirit, one might also ask if there are obstructions to embedding many disjoint (integral) Lagrangian tori in a fixed symplectic manifold of finite volume. In this talk I will discuss two results in this direction. The first asserts that the Clifford torus in $S^2 \times S^2$ is a maximal Lagrangian packing in the sense that any other integral Lagrangian torus must intersect it. The second result shows that a natural candidate for a maximal symplectic packing of a polydisc actually fails to be maximal. This is joint work with Richard Hind.

5:00 pm in 241 Altgeld Hall,Monday, September 16, 2019

#### Stability property of conditional probabilities

###### Chris Linden (UIUC)

Abstract: We will present the second part on different forms of equivalences for conditional probabilities developed in the IGL project last semester.

Tuesday, September 17, 2019

12:00 pm in 243 Altgeld Hall,Tuesday, September 17, 2019

#### Coloring invariants of knots and links are often intractable

###### Eric Sampterton (Illinois Math)

Abstract: I’ll give an overview of my result with Greg Kuperberg concerning the computational complexity of G-coloring invariants of knots, where G is a finite, simple group. We have a similar theorem for closed 3-manifolds. I’ll try to give a sense of the commonalities of the two proofs (e.g. “reversible computing with a combinatorial TQFT”), as well as where they differ (there’s some interesting algebraic topology that needed developing in the knot case). Time permitting, I’ll discuss the special case of hyperbolic knots and 3-manifolds.

1:00 pm in 345 Altgeld Hall,Tuesday, September 17, 2019

#### Conjugacy classes of automorphism groups of linearly ordered structures

###### Aleksandra Kwiatkowska (Universität Münster and Uniwersytet Wrocławski)

Abstract: In the talk, we will address the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a countable structure or of a Fraisse limit) that is extremely amenable and has ample generics. In fact, it is unknown if there exists a linearly ordered structure whose automorphism group has a comeager $2$-dimensional diagonal conjugacy class.
We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager 2-dimensional diagonal conjugacy class. Moreover, we provide general conditions implying that there is no comeager conjugacy class or comeager $2$-dimensional diagonal conjugacy class in the automorphism group of an ordered Fraisse limit.
This is joint work with Maciej Malicki.

1:00 pm in 347 Altgeld Hall,Tuesday, September 17, 2019

#### Finite Elements for Curvature

###### Kaibo Hu   [email] (University of Minnesota)

Abstract: We review the elasticity (linearized Calabi) complex, its cohomology and potential applications in differential geometry and continuum defect theory. We construct discrete finite element complexes. In particular, this leads to new finite element discretization for the 2D linearized curvature operator. Compared with classical discrete geometric approaches, e.g., the Regge calculus, the new finite elements are conforming. The construction is based on a Bernstein-Gelfand-Gelfand type diagram chase with various finite element de Rham complexes. This is a joint work with Snorre H. Christiansen.

2:00 pm in 243 Altgeld Hall,Tuesday, September 17, 2019

#### Connected Fair Detachments of Hypergraphs

###### Amin Bahmanian (ISU Math)

Abstract: Let $\mathcal G$ be a hypergraph whose edges are colored. An $(\alpha,n)$-detachment of $\mathcal G$ is a hypergraph obtained by splitting a vertex $\alpha$ into $n$ vertices, say $\alpha_1,\dots,\alpha_n$, and sharing the incident hinges and edges among the subvertices. A detachment is fair if the degree of vertices and multiplicity of edges are shared as evenly as possible among the subvertices within the whole hypergraph as well as within each color class. We find necessary and sufficient conditions under which a $k$-edge-colored hypergraph $\mathcal G$ has a fair detachment in which each color class is connected. Previously, this was not even known for the case when $\mathcal G$ is an arbitrary graph (i.e. 2-uniform hypergraph). We exhibit the usefulness of our theorem by proving a variety of new results on hypergraph decompositions, and completing partial regular combinatorial structures.

3:00 pm in 243 Altgeld Hall,Tuesday, September 17, 2019

#### P=W, a strange identity for Hitchin systems

###### Zili Zhang (U Michigan)

Abstract: Start with a compact Riemann surface X with marked points and a complex reductive group G. According to Hitchin-Simpson’s nonabelian Hodge theory, the pair (X,G) comes with two new complex varieties: the character variety M_B and the Higgs moduli M_D. I will present some aspects of this story and discuss an identity P=W indexed by affine Dynkin diagrams – occurring in the singular cohomology groups of M_D and M_B, where P and W dwell. Based on joint work with Junliang Shen.

Wednesday, September 18, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, September 18, 2019

#### Polish groups whose measure preserving actions are whirly

###### Pavlos Motakis (UIUC Math)

Abstract: Let $\mathrm{MALG}(X)$ denote the measure algebra of a standard probability space $(X,\mu)$. A measure preserving action of a Polish group $G$ on $\mathrm{MALG}(X)$ is called whirly if for any $A, B$ in $\mathrm{MALG}(X)$ with positive measure and for any open neighborhood $U$ of the identity of $G$ there exists $g\in U$ so that $(gA)\cap B$ has positive measure. We follow two papers, one of Glasner–Tsirelson–Weiss and one of Glasner–Weiss, to prove two results. The first one is that if $G$ is certain type of Polish group, namely a Lévy group, then any non-trivial Borel action on $\mathrm{MALG}(X)$ is whirly. The second result is that for the generic automorphism $T\in\mathrm{MALG}(X)$, the closed subgroup of automorphisms generated by $T$ acts on $\mathrm{MALG}(X)$ whirlily.
This is a follow up to the lecture of D. Ihli on 09/11/2019.

4:00 pm in 447 Altgeld Hall,Wednesday, September 18, 2019

#### Intro to the Gorsky-Negut wall-crossing conjecture

###### Josh Wen (Illinois Math)

Abstract: The Hilbert scheme of points on the plane is a space that by now has been connected to many areas outside of algebraic geometry: e.g. algebraic combinatorics, representation theory, knot theory, etc. The equivariant K-theory of these spaces have a few distinguished bases important to making some of these connections. A new entrant to this list of bases is the Maulik-Okounkov K-theoretic stable bases. They depend in a piece-wise constant manner by a real number called the slope, and the numbers where the bases differ are called the walls. Gorsky and Negut have a conjecture relating the transition between bases when the slope crosses a wall to the combinatorics of q-Fock spaces for quantum affine algebras. I'll try to introduce as many of the characters of this story as I can as well as discuss a larger picture wherein these stable bases are geometric shadows of things coming from deformation quantization.

Thursday, September 19, 2019

11:00 am in 241 Altgeld Hall,Thursday, September 19, 2019

#### Indivisibility and divisibility of class numbers of imaginary quadratic fields

###### Olivia Beckwith (Illinois)

Abstract: For any prime p > 3, the strongest lower bounds for the number of imaginary quadratic fields with discriminant down to -X for which the class group has trivial (resp. non-trivial) p-torsion are due to Kohnen and Ono (Soundararajan). I will discuss refinements of these classic results in which we consider the imaginary quadratic fields for which the class number is indivisible (divisible) by p and which satisfy the property that a given finite set of rational primes split in a prescribed way. We prove a lower bound for the number of such fields with discriminant down to -X which is of the same order of magnitude as in Kohnen and Ono's (Soundararajan's) results. For the indivisibility case, we rely on a result of Wiles establishing the existence of imaginary quadratic fields with trivial p-torsion in their class groups which satisfy a finite set of local conditions, and a result of Zagier which says that the Hurwitz class numbers are the Fourier coefficients of a mock modular form.

1:00 pm in 464 Loomis,Thursday, September 19, 2019

#### QUANTUM GRAVITY AND THE SWAMPLAND

###### Gary Shiu (University of Wisconsin)

Abstract: String theory seems to offer an enormous number of possibilities for low energy physics. The huge set of solutions is often known as the String Theory Landscape. In recent years, however, it has become clear that not all quantum field theories can be consistently coupled to gravity. Theories that cannot be ultraviolet completed in quantum gravity are said to be in the Swampland. In this talk, I’ll discuss some conjectured properties of quantum gravity, evidences for them, and their applications to cosmology.

2:00 pm in 243 Altgeld Hall,Thursday, September 19, 2019

#### Inductive limits of C*-algebras and compact quantum metric spaces

###### Konrad Aguilar (Arizona State University)

Abstract: In this talk, we will place quantum metrics, in the sense of Rieffel, on certain unital inductive limits of C*-algebras built from quantum metrics on the terms of the given inductive sequence with certain compatibility conditions. One of these conditions is that the inductive sequence forms a Cauchy sequence of quantum metric spaces in the dual Gromov-Hausdorff propinquity of Latremoliere. Since the dual propinquity is complete, this will produce a limit quantum metric space. Based on our assumptions, we then show that the C*-algebra of this limit quantum metric space is isomorphic to the given inductive limit, which finally places a quantum metric on the inductive limit. This then immediately allows us to establish a metric convergence of the inductive sequence to the inductive limit. Another consequence to our construction is that we place new quantum metrics on all unital AF algebras that extend our previous work with Latremoliere on unital AF algebras with faithful tracial state.

2:00 pm in 347 Altgeld Hall,Thursday, September 19, 2019

#### The Yang Mills Problem for Probabilists

###### Kesav Krishnan (UIUC math)

Abstract: We aim to introduce the problem of rigorously defining the Yang-Mills field from the probability perspective. In this first talk, we will introduce lattice guage theory, and some geometric preliminaries

4:00 pm in 245 Altgeld Hall,Thursday, September 19, 2019

#### On the container method

###### Jozsef Balogh   [email] (University of Illinois at Urbana-Champaign)

Abstract: We will give a gentle introduction to a recently-developed technique, `The Container Method’, for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of 'containers' for the independent sets, each of which contains few edges. The container method is very useful counting discrete structures with certain properties; transferring theorems into random environment; and proving the existence discrete structures satisfying some important properties. In the first half of the talk we will attempt to convey a general high-level overview of the method, in particular how independent sets in hypergraphs could be used to model various problems in combinatorics; in the second, we will describe a few illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry.

Friday, September 20, 2019

2:00 pm in 147 Altgeld Hall,Friday, September 20, 2019

#### Escaping nontangentiality: Towards a controlled tangential amortized Julia-Carathéodory theory

###### Meredith Sargent (University of Arkansas)

Abstract: Let $f: D \rightarrow \Omega$ be a complex analytic function. The Julia quotient is given by the ratio between the distance of $f(z)$ to the boundary of $\Omega$ and the distance of $z$ to the boundary of $D.$ A classical Julia-Carathéodory type theorem states that if there is a sequence tending to $\tau$ in the boundary of $D$ along which the Julia quotient is bounded, then the function $f$ can be extended to $\tau$ such that $f$ is nontangentially continuous and differentiable at $\tau$ and $f(\tau)$ is in the boundary of $\Omega.$ We develop an extended theory when $D$ and $\Omega$ are taken to be the upper half plane which corresponds to amortized boundedness of the Julia quotient on sets of controlled tangential approach, so-called $\lambda$-Stolz regions, and higher order regularity, including but not limited to higher order differentiability, which we measure using $\gamma$-regularity. I will discuss the proof, along with some applications, including moment theory and the fractional Laplacian. This is joint work with J.E. Pascoe and Ryan Tully-Doyle.

2:00 pm in 347 Altgeld Hall,Friday, September 20, 2019

#### Hopf Ore Extensions

###### Hongdi Huang (University of Waterloo (visiting UIUC for F19))

Abstract: Brown, O'Hagan, Zhang, and Zhuang gave a set of conditions on an automorphism $\sigma$ and a $\sigma$-derivation $\delta$ of a Hopf $k$-algebra $R$ for when the skew polynomial extension $T=R[x, \sigma, \delta]$ of $R$ admits a Hopf algebra structure that is compatible with that of $R$. In fact, they gave a complete characterization of which $\sigma$ and $\delta$ can occur under the hypothesis that $\Delta(x)=a\otimes x +x\otimes b +v(x\otimes x) +w$, with $a, b\in R$ and $v, w\in R\otimes_k R$, where $\Delta: R\to R\otimes_k R$ is the comultiplication map. In this paper, we show that after a change of variables one can in fact assume that $\Delta(x)=\beta^{-1}\otimes x +x\otimes 1 +w$, with $\beta$ is a grouplike element in $R$ and $w\in R\otimes_k R,$ when $R\otimes_k R$ is a domain and $R$ is noetherian. In particular, this completely characterizes skew polynomial extensions of a Hopf algebra that admit a Hopf structure extending that of the ring of coefficients under these hypotheses. We show that the hypotheses hold for domains $R$ that are noetherian cocommutative Hopf algebras of finite Gelfand-Kirillov dimension.

3:00 pm in 341 Altgeld Hall,Friday, September 20, 2019

#### Introduction to Quasiconformal and Quasisymmetric maps on metric spaces

###### Stathis Chrontsios (UIUC Math)

Abstract: The talk will be a quick introduction to quasiconformal and quasisymmetric maps on metric spaces. I will start by describing how quasiconformal maps first appeared as generalizations of conformal maps on the complex plane and how they were generalized in arbitrary metric spaces. In addition, I will present how they gave rise to quasisymmetric maps on the real line and their later generalization in metric spaces. Moreover, I will discuss interesting quasisymmetric invariants and the definition of the conformal gauge. Last but not least, I will mention some applications this theory has had in Geometric Group Theory and some open problems.

3:00 pm in 1 Illini Hall,Friday, September 20, 2019

#### The prime number theorem through the Ingham-Karamata Tauberian theorem

###### Gregory Debruyne (Illinois Math)

Abstract: It is well-known that the prime number theorem can be deduced from certain Tauberian theorems. In this talk, we shall present a Tauberian approach that is perhaps not that well-known through the Ingham-Karamata theorem. Moreover, we will give a recently discovered "simple" proof of a so-called one-sided version of this theorem. We will also discuss some recent developments related to the Ingham-Karamata theorem. The talk is based on work in collaboration with Jasson Vindas.

4:00 pm in 345 Altgeld Hall,Friday, September 20, 2019

#### A Logician's Introduction to the Problem of P vs. NP

###### Alexi Block Gorman (UIUC Math)

Abstract: Central to much of computer science, and some areas of mathematics, are questions about various problems' computability and complexity (whether the problem can be solved "algorithmically," and how "hard" it is to do so). In this talk, I will first give an overview of the complexity hierarchy for machines (from finite automata to Turing machines) and the mathematical properties of the space of languages that we associate with them. Next, I will discuss the relationship of deterministic and non-deterministic machines, which will allow us to segue from questions of computability to that of complexity. Finally, I will give a precise formulation of the problem of P vs. NP, and try to illustrate why the problem remains rather elusive. This talk does not require any background in logic or computer science, and should be accessible to all graduate students.

4:00 pm in 347 Altgeld Hall,Friday, September 20, 2019

#### To Be Announced

###### Alice Chudnovsky (UIUC Math)

4:00 pm in 141 Altgeld Hall,Friday, September 20, 2019

#### A Geometric Proof of Lie's Third Theorem

###### Shuyu Xiao (UIUC)

Abstract: There are three basic results in Lie theory known as Lie's three theorems. These theorems together tell us that: up to isomorphism, there is a one-to-one correspondence between finite-dimensional Lie algebras and simply connected Lie groups. While the first two theorems are easy to prove with the most basic differential geometry knowledge, the third one is somehow a deeper result which needs relatively advanced tools. In this talk, I will go over the proof given by Van Est, in which he identifies any finite-dimensional Lie algebra with a semi-direct product of its center and its adjoint Lie algebra. I will introduce Lie group cohomology, Lie algebra cohomology and how they classify the abelian extensions of Lie groups and Lie algebras and thus determine the Lie algebra structure on the semi-direct product.