Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, February 20, 2018.

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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.