Department of

# Mathematics

Seminar Calendar
for Descriptive Set Theory Seminar events the year of Thursday, July 2, 2020.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
      June 2020              July 2020             August 2020
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1  2  3  4  5  6             1  2  3  4                      1
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30 31


Wednesday, January 22, 2020

3:30 pm in 341 Altgeld Hall,Wednesday, January 22, 2020

#### Organizational meeting

Wednesday, January 29, 2020

3:30 pm in 341 Altgeld Hall,Wednesday, January 29, 2020

#### Introduction to IRS

###### Jenna Zomback and Anush Tserunyan

Abstract: This is an introductory talk on Invariant Random Subgroups (IRS), which can be viewed as probabilistic generalization of normal subgroups and lattices. We will show that for all countable groups, all IRS arise from pmp actions, and discuss Kesten's theorem for IRS. All this is from the paper "Kesten's theorem for Invariant Random Subgroups" by Abert, Glasner, and Virag [arXiv].

Wednesday, February 5, 2020

3:30 pm in 341 Altgeld Hall,Wednesday, February 5, 2020

#### Cancelled

Wednesday, February 12, 2020

3:30 pm in 341 Altgeld Hall,Wednesday, February 12, 2020

#### Strongly amenable groups

###### Joshua Frisch (Caltech Math)

Abstract: A topological dynamical system (i.e. a group acting by homeomorphisms on a compact Hausdorff space) is said to be proximal if for any two points $p$ and $q$ we can simultaneously "push them together" (rigorously, there is a net $g_n$ such that $\lim g_n(p) = \lim g_n(q)$). In his paper introducing the concept of proximality, Glasner noted that whenever $\mathbb{Z}$ acts proximally, that action will have a fixed point. He termed groups with this fixed point property "strongly amenable" and showed that non-amenable groups are not strongly amenable and virtually nilpotent groups are strongly amenable. In this talk I will discuss recent work precisely characterizing which (countable) groups are strongly amenable. This is joint work with Omer Tamuz and Pooya Vahidi Ferdowsi.

Wednesday, March 11, 2020

3:30 pm in 341 Altgeld Hall,Wednesday, March 11, 2020

#### Random walks on graphs and spectral radius: part 1

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

Abstract: To motivate Kesten's theorem and its version for IRS, we will discuss random walks on graphs, the associated Markov operators, and the spectral radius. We will prove that the spectral radius is equal to the norm of the Markov operator.

Wednesday, April 1, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for the password),Wednesday, April 1, 2020

#### Random walks on graphs and spectral radius: part 2

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

Abstract: We will continue discussing random walks on graphs and their spectral radius, computing that the spectral radius of $\mathbb{Z}^d$ is $1$, whereas it is less than $1$ for a $d$-regular tree with $d \ge 3$. We will then discuss the deep and general theorem of Kesten characterizing amenable normal subgroups and derive a couple of striking corollaries: a characterization of finitely generated amenable groups and a rigidity result for the free groups.

Wednesday, April 8, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for the password),Wednesday, April 8, 2020

#### Random walks on graphs and spectral radius: part 3

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

Abstract: We will continue discussing random walks on graphs. In this last talk of the series, we will consider recurrence/transience of random walks, proving that this is determined by whether or not the expectation of the number of visits to a fixed vertex is infinite. We will use it to deduce that the simple random walk on nonamenable Cayley graphs is transient. We will also show that the simple random walk on $\mathbb{Z}^d$ is recurrent if and only if $d \le 2$. As Kakutani put it, "A drunk man will find his way home, but a drunk bird may get lost forever."

Wednesday, April 15, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for password),Wednesday, April 15, 2020

#### An obstruction for classification by actions of TSI Polish groups, part 2: proofs

###### Aristotelis Panagiotopoulos (Caltech Math)

Abstract: In this talk we will over the proof of my recent result (joint with Shaun Allison) that if a $G$-space $X$ is generically unbalanced then its orbit equivalence relation is not classifiable by actions of TSI Polish groups. I will also discuss how one can use this result to show that Morita equivalence between continuous-trace $C^*$ algebras, as well as isomorphism between Hermitian line bundles, are not classifiable by TSI group actions.

Wednesday, April 22, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for password),Wednesday, April 22, 2020

#### Introduction to 𝓁2-Betti numbers for groups, part 1: Homology

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

Wednesday, April 29, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for password),Wednesday, April 29, 2020

#### Descriptive combinatorics, distributed algorithms, and the Lovász Local Lemma: proofs

###### Anton Bernshteyn (CMU Math)

Abstract: Descriptive combinatorics is the study of combinatorial problems (such as graph coloring) under additional topological or measure-theoretic regularity restrictions. It turns out that there is a close relationship between descriptive combinatorics and distributed computing, i.e., the area of computer science concerned with problems that can be solved efficiently by a decentralized network of processors. At the heart of this relationship lies the Lovász Local Lemma—an important tool in probabilistic combinatorics—and its measurable versions. In this talk I will sketch the arguments behind this relationship.

Wednesday, May 6, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for password),Wednesday, May 6, 2020

#### Introduction to 𝓁2-Betti numbers for groups, part 2: Group homology

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

Abstract: We will discuss equivariant homology for spaces equipped with a group action, defined as the ordinary homology of the "homotopy quotient" of the action. Group homology is the special case for the trivial action on a point. Time permitting, we will then begin to discuss the theory of (Hattori-Stallings) traces, which provides a better-behaved equivariant analog of torsion-free rank.

Wednesday, May 13, 2020

3:30 pm in https://illinois.zoom.us/j/806582029 (email Anush Tserunyan for password),Wednesday, May 13, 2020

#### Introduction to 𝓁2-Betti numbers for groups, part 3: 𝓁2-homology

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

Abstract: We will first finish our discussion of ordinary group homology by giving an alternative definition via equivariant chain complexes, completely bypassing topology. We will then discuss $\ell^2$-homology, defined by replacing (chain) homotopy quotients by $\ell^2$-completion to Hilbert $\Gamma$-modules. The $\ell^2$-Betti numbers are the von Neumann dimensions of the resulting homology Hilbert $\Gamma$-modules.