Department of

# Mathematics

Seminar Calendar
for events the week of Wednesday, February 21, 2018.

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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, February 19, 2018

3:00 pm in 243 Altgeld Hall,Monday, February 19, 2018

#### Symplectic groupoids and monoidal Fukaya categories

###### James Pascaleff (UIUC)

Abstract: I will describe how a groupoid structure on a symplectic manifold naturally induces a monoidal structure on its Fukaya category. This provides a unifying perspective on the various known monoidal structures on Fukaya categories. As an application, I will use this framework to address the question of when Lagrangian Floer cohomology rings are commutative.

3:00 pm in Altgeld Hall 345,Monday, February 19, 2018

#### Why should homotopy theorists care about homotopy type theory?

###### Nima Rasekh (UIUC Math)

Abstract: There is this new branch of mathematics known as homotopy type theory, which combines concepts from logic, computer science and homotopy theory. While it is clearly not necessary to know homotopy type theory to learn homotopy theory there are certain aspects that can be beneficial. The goal of this talk is to point to some of those aspects of homotopy type theory that can be beneficial to our understanding of homotopy theory.

4:00 pm in 245 Altgeld Hall,Monday, February 19, 2018

#### Initial and boundary value problems for dispersive partial differential equations

###### Nikos Tzirakis   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: In this talk we will introduce some basic methods based on Fourier transform techniques to obtain solutions for some nonlinear dispersive partial differential equations that are posed on an infinite or a semi-infinite domain. Examples include the Korteweg de Vries equation (KdV) and the nonlinear Schrodinger equation (NLS).

5:00 pm in 241 Altgeld Hall,Monday, February 19, 2018

#### Modules and Cost

###### Marius Junge and Anush Tserunyan

Abstract: We will present some examples of Hilbert W*-modules and start with the cost of an equivalence relation.

Tuesday, February 20, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, February 20, 2018

#### Which groups have bounded harmonic functions?

###### Yair Hartman (Northwestern University)

Abstract: Bounded harmonic functions on groups are closely related to random walks on groups. It has long been known that all abelian groups, and more generally, virtually nilpotent groups are "Choquet-Deny groups": these groups cannot support non-trivial bounded harmonic functions. Equivalently, their Furstenberg-Poisson boundary is trivial, for any random walk. I will present a very recent result where we complete the classification of discrete countable Choquet-Deny groups. In particular, we show that any finitely generated group which is not virtually nilpotent, is not Choquet-Deny. Surprisingly, the key is not the growth rate of the group, but rather the algebraic infinite conjugacy class property (ICC). This is joint work with Joshua Frisch, Omer Tamuz and Pooya Vahidi Ferdowsi.

1:00 pm in 345 Altgeld Hall,Tuesday, February 20, 2018

#### A Descriptive Set in Topological Dynamics

###### Robert Kaufman (Illinois Math)

Abstract: Let $X$ be a compact metric space and $H(X)$ the group of homeomorphisms of $X$, a Polish group. Then the orbit of $x \in X$ under $h \in H(X)$ (the two-sided orbit) has an obvious definition; its closure is called the orbit-closure of $x$. When all the orbit-closures are minimal then $h$ is called "sharp". (Every orbit-closure contains a minimal one). There are two main theorems.
A. The set $S(X)$ of sharp homeomorphisms is always co-analytic.
B. For a certain $X$, $S(X)$ is not Borel.
In the proof of B we need a variant of Hurewicz' theorem (1930) on the class of uncountable compact sets. This variant should be (but so far isn't) a consequence of Hurewicz' theorem. I'll say a very few words about a new method of proof.

3:00 pm in 241 Altgeld Hall,Tuesday, February 20, 2018

#### Fractional DP-Colorings

###### Anton Bernshteyn (Illinois Math)

Abstract: DP-coloring is a generalization of list coloring introduced by Dvořák and Postle in 2015. This talk will be about a fractional version of DP-coloring. There is a natural way to define fractional list coloring; however, Alon, Tuza, and Voigt proved that the fractional list chromatic number of any graph coincides with its ordinary fractional chromatic number. This result does not extend to fractional DP-coloring: The difference between the fractional DP-chromatic number and the ordinary fractional chromatic number of a graph can be arbitrarily large. A somewhat surprising fact about DP-coloring is that the DP-chromatic number of a triangle-free regular graph is essentially determined by its degree. It turns out that for fractional DP-coloring, this phenomenon extends to a much wider class of graphs (including all bipartite graphs, for example). This is joint work with Alexandr Kostochka (UIUC) and Xuding Zhu (Zhejiang Normal University).

4:00 pm in Illini Hall 1,Tuesday, February 20, 2018

#### A Bird's-Eye View of Seiberg Witten Integrable Systems

###### Matej Penciak (UIUC)

Abstract: In this talk I will give a rudimentary description of supersymmetric gauge theories, and focus on the particular case of $N=2$ supersymmetry in dimension $4$ with gauge group $SU(2)$. In this setting, originally noticed and explained by Seiberg and Witten in 1994, the moduli of vacua exhibits the structure of an algebraic integrable system. I will explain how this structure manifests itself, and the give a sketch of the calculation that Seiberg and Witten made in their original paper. If time permits, I will explain the generalization of this story to more general gauge groups, and with possible additional matter fields included in the theory.

4:00 pm in 243 Altgeld Hall,Tuesday, February 20, 2018

#### A pointwise ergodic theorem for quasi-pmp graphs

###### Anush Tserunyan (UIUC Math)

Abstract: We prove a pointwise ergodic theorem for quasi-pmp locally countable graphs, which states that the global condition of ergodicity amounts to locally approximating the means of $L^1$-functions via increasing subgraphs with finite connected components. The pmp version of this theorem was first proven by R. Tucker-Drob using probabilistic methods. Our proof is different: it is constructive and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant and a simple method of exploiting nonamenability. The non-pmp setting additionally requires a new gadget for analyzing the interplay between the underlying cocycle and the graph.

Wednesday, February 21, 2018

12:00 pm in 443 Altgeld Hall,Wednesday, February 21, 2018

#### Bounding Betti numbers of patchworked real hypersurfaces by Hodge numbers

###### Kristin Shaw (MPI Leipzig)

Abstract: The Smith-Thom inequality bounds the sum of the Betti numbers of a real algebraic variety by the sum of the Betti numbers of its complexification. In this talk I will explain our proof of a conjecture of Itenberg which refines this bound for a particular class of real algebraic projective hypersurfaces in terms of the Hodge numbers of its complexification. The real hypersurfaces we consider arise from Viro’s patchworking construction, which is a powerful combinatorial method for constructing topological types of real algebraic varieties. To prove the bounds conjectured by Itenberg, we develop a real analogue of tropical homology and use spectral sequences to compare it to the usual tropical homology of Itenberg, Katzarkov, Mikhalkin, Zharkov. Their homology theory gives the Hodge numbers of a complex projective variety from its tropicalisation. Lurking in the spectral sequences of the proof are the keys to controlling the topology of the real hypersurface produced from a patchwork. This is joint work in preparation with Arthur Renaudineau.

Thursday, February 22, 2018

11:00 am in 241 Altgeld Hall,Thursday, February 22, 2018

#### Euler Systems and Special Values of L-functions

###### Corey Stone (University of Illinois)

Abstract: In the 1990s, Kolyvagin and Rubin introduced the Euler system of Gauss sums to derive upper bounds on the sizes of the p-primary parts of the ideal class groups of certain cyclotomic ﬁelds. Since then, this and other Euler systems have been studied in order to analyze other number-theoretic structures. Recent work has shown that Kolyvagin’s Euler system appears naturally in the context of various conjectures by Gross, Rubin, and Stark involving special values of L-functions. We will discuss these Euler systems from this new point of view as well as a related result about the module structure of various ideal class groups over Iwasawa algebras.

2:00 pm in 241 Altgeld Hall,Thursday, February 22, 2018

#### Multiples of long period small element continued fractions to short period large elements continued fractions

###### Michael Oyengo (UIUC)

Abstract: We construct a class of rationals and quadratic irrationals having continued fractions whose period has length $n\geq2$, and with "small'' partial quotients for which certain integer multiples have continued fractions of period $1$, $2$ or $4$ with "large'' partial quotients. We then show that numbers in the period of the new continued fraction are simple functions of the numbers in the periods of the original continued fraction. We give generalizations of these continued fractions and study properties of polynomials arising from these generalizations.

2:00 pm in 243 Altgeld Hall,Thursday, February 22, 2018

#### Basic, order basic, and bibasic sequences in Banach lattices

###### Vladimir Troitsky (University of Alberta)

Abstract: Recall that sequence in a Banach is a (Schauder) basis if every vector admits a unique series expansion which converges to the vector in norm. In a Banach lattice, one may replace norm convergence with order or uniform convergence. This leads to several types of order bases and order basic sequences. We will discuss connections between these types of sequences. This is a joint project with M.Taylor.

4:00 pm in 245 Altgeld Hall,Thursday, February 22, 2018

###### Sergey Norin (McGill University)

Abstract: Hadwiger's conjecture from 1943 states that every simple graph with no $K_t$ minor can be properly colored using t-1 colors. This is a far-reaching strengthening of the four-color theorem and appears to be currently out of reach in its full generality. In the last three years, however, several relaxations have been proven. In these relaxations one considers colorings such that every color class induces a subgraph with bounded maximum degree or with bounded component size. We will survey recent results on such improper colorings of minor-closed classes of graphs. Based on joint work with Zdenek Dvorak and with Alex Scott, Paul Seymour and David Wood.

Friday, February 23, 2018

4:00 pm in 345 Altgeld Hall,Friday, February 23, 2018

#### "On dp-minimal ordered structures" by P. Simon (part 1)

###### Travis Nell (Illinois Math)

Abstract: This is an introductory talk to "On dp-minimal ordered structures" by P. Simon [arXiv].

4:00 pm in 241 Altgeld Hall,Friday, February 23, 2018

#### Unoriented Bordism and the Algebraic Geometry of Homotopy Theory

###### Brian Shin (UIUC)

Abstract: The classification of manifolds up to bordism is an interesting geometric problem. In this talk, I will discuss how this problem connects to homotopy theory. In particular, I will demonstrate that viewing the homotopy theorist's toolbox through the lens of algebraic geometry leads naturally to the solution for the classification of manifolds up to bordism.