Department of

Mathematics


Seminar Calendar
for Descriptive Set Theory Seminar events the year of Monday, October 26, 2020.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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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.

Friday, September 25, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, September 25, 2020

Characterizing companionability for expansions of o-minimal theories by a dense, proper subgroup

Alexi Block Gorman (UIUC Math)

Abstract: Recent works in model theory have established natural and broad criteria concerning the existence of model companions and the preservation of certain neostability properties when passing to the model companion. In this talk, we restrict our attention to the o-minimal setting. By doing so, we can isolate the sort of necessary and sufficient condition that can be elusive in more general settings. The central result is a full characterization for when the expansion of a complete o-minimal theory by a unary predicate that picks out a dense, divisible subgroup has a model companion. We will discuss examples both in which the predicate is an additive subgroup, and in which it is a mutliplicative subgroup. The o-minimal setting allows us to provide a full and geometric characterization for companionability, with a particularly elegant dividing line when the group operation is multiplication. We conclude with a brief discussion of neostability properties, and give examples that illustrate the lack of preservation for properties such as strong, NIP, and NTP2, though there are also examples for which some or all three of those properties hold.

Friday, October 2, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, October 2, 2020

T-convex T-differential fields and their immediate extensions (Part 1)

Elliot Kaplan (UIUC Math)

Abstract: Let T be an o-minimal theory extending the theory of ordered fields. A T-convex T-differential field is a model of T equipped with a T-convex valuation ring and a continuous T-derivation. This week and next week, I will discuss some of my recent work on immediate extensions of T-convex T-differential fields. This week will be focused on background (what a T-convex valuation ring is, what a T-derivation is, what immediate extensions are) and on examples of T-convex T-differential fields.

Friday, October 9, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, October 9, 2020

T-convex T-differential fields and their immediate extensions (Part 2)

Elliot Kaplan (UIUC Math)

Abstract: Continuing from last week, I will discuss T-derivations and strict extensions, define "eventual smallness", and sketch a proof of the main result: if T is polynomially bounded, then any T-convex T-differential field has an immediate strict extension which is spherically complete.

Friday, October 23, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, October 23, 2020

On the Pila-Wilkie Theorem

Neer Bhardwaj (UIUC Math)

Abstract: We prove Pila and Wilkie’s Counting theorem, following the original paper, but exploit cell decomposition more thoroughly to simplify the deduction from its main ingredients. Our approach in particular completely avoids ‘regular’ or $C^1$ smooth points, and related technology; which also allows simplifications around Pila’s ‘block family’ refinement of the result.

Friday, October 30, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, October 30, 2020

What generic automorphisms of the random poset look like

Dakota Thor Ihli (UIUC Math)

Abstract: The Fraïssé limit of the class of finite posets, also called the random poset, admits generic automorphisms — that is, its automorphism group admits a comeagre conjugacy class. This result, due to D. Kuske and J. Truss, was proven without explicitly describing the automorphisms in question. Here we give a new, concrete description of the generic automorphisms, and we discuss the tools-and-tricks involved.

Friday, November 6, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, November 6, 2020

The fractal properties of real sets defined by Büchi automata

Christian Schulz (UIUC Math)

Abstract: Büchi automata are a natural counterpart to finite automata that accept infinite strings instead of finite strings as input. We consider the k-automatic sets, which are subsets of $[0, 1]^n$ consisting of all tuples with a base-k expansion recognized by a given Büchi automaton. These sets often exhibit fractal behavior; for instance, the Cantor set and Sierpinski carpet are both 3-automatic. In this talk, we give methods for measuring various fractal properties of a k-automatic set, including box-counting dimension and Hausdorff dimension and measure, by examining the structure of its Büchi automaton. This is joint work with Alexi Block Gorman and Philipp Hieronymi.

Friday, November 13, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, November 13, 2020

GraphSAT -- a novel decision problem combining SAT and graph theory

Vaibhav Karve (UIUC Math)

Abstract: Graph theorists often care about forbidden-graph characterizations of graph properties. For example, Kuratowski's theorem says that a graph is planar iff it does not "contain" K5 or K3,3. We replace "planar" by a new graph property inspired by Boolean Satisfiability (SAT) and look for similar forbidden graphs in the hope that it will tell us something about SAT. We will also talk about various ways to translate "containment" to GraphSAT and as a result we will encounter a host of other decision problems on the way. We will show a full forbidden-graph characterization of 2-GraphSAT and partial results for 3-GraphSAT. This talk requires no prerequisites other than familiarity with complexity classes P, NP, NP-complete etc. This is joint work with Anil Hirani (UIUC Math)

Friday, November 20, 2020

4:00 pm in Zoom (email ruiyuan at illinois for info),Friday, November 20, 2020

A universal separable homogeneous Banach lattice

Mary Angelica Tursi (UIUC Math)

Abstract: A metric structure $\mathfrak{M}$ is considered approximately ultra-homogeneous if for any finitely generated structure $A =\langle a_1,\dotsc,a_n\rangle$, embeddings $f,g:A \rightarrow \mathfrak{M}$, and $\varepsilon > 0$, there exists an automorphism $\phi$ on $\mathfrak{M}$ such that $d(\phi \circ f (a_i), g(a_i) ) < \varepsilon $. We show that there exists a separable approximately ultra-homogeneous Banach lattice $\mathfrak{BL}$ that is isometrically universal for separable Banach lattices by proving that finitely generated Banach lattices form a metric Fraïssé class. If time permits, we will also explore some interesting properties about $\mathfrak{BL}$.