Department of

# Mathematics

Seminar Calendar
for Descriptive Set Theory Seminar events the year of Saturday, February 29, 2020.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     January 2020          February 2020            March 2020
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4                      1    1  2  3  4  5  6  7
5  6  7  8  9 10 11    2  3  4  5  6  7  8    8  9 10 11 12 13 14
12 13 14 15 16 17 18    9 10 11 12 13 14 15   15 16 17 18 19 20 21
19 20 21 22 23 24 25   16 17 18 19 20 21 22   22 23 24 25 26 27 28
26 27 28 29 30 31      23 24 25 26 27 28 29   29 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,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 consider another parameter used in studying a given random walk, namely, the expectation of the number of visits to a fixed vertex. This controls the recurrence of the random walk and we will use it to deduce that the simple random walk on $\mathbb{Z}^d$ is recurrent if and only if $d \le 2$.

Wednesday, April 22, 2020

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