Anton Bernshteyn, Complexity classes of local problems
Abstract: In this talk I will discuss the complexity hierarchy for locally checkable labeling problems, a.k.a. subshifts of finite type, based on concepts from descriptive set theory, probability theory, and computability theory. The main focus of the talk will be on the free group $\mathbb{F}_n$ and the free Abelian group $\mathbb{Z}^n$ and the striking differences between them. This is joint work with Katalin Berlow, Clark Lyons, and Felix Weilacher.
Erin Bevilacqua, A new criteria for maximal cost
Abstract: Recent works have expanded the idea of cost to locally compact groups using the maximality of the cost of Poisson point processes. In this talk we return this concept to the discrete (countable) case. Abert-Weiss showed in 2013 that cost is maximal for essentially free pmp actions which are weakly Bernoulli. We combine these ideas by showing that the Poisson suspension of any weakly amenable infinite measure preserving (imp) action is weakly Bernoulli and hence maximal, and that the cost of a Poisson suspension can be bounded above by the normalized cost of the action when the imp is doubly recurrent. We hope to apply this idea to bound the cost of classes of groups which are not known to have fixed price. This is a work in progress and is joint with Lewis Bowen.
Michael Chapman, Subgroup Tests and the Aldous-Lyons conjecture
Abstract: The Aldous-Lyons conjecture from probability theory states that every (unimodular random) infinite graph can be (Benjamini-Schramm) approximated by finite graphs. This conjecture is an analogue of other influential conjectures in mathematics concerning how well certain infinite objects can be approximated by finite ones; examples include Connes' embedding problem (CEP) in functional analysis and the soficity problem of Gromov-Weiss in group theory. These became major open problems in their respective fields, as many other long standing open problems, that seem unrelated to any approximation property, were shown to be true for the class of finitely-approximated objects. For example, Gottschalk's conjecture and Kaplansky's direct finiteness conjecture are known to be true for sofic groups, but are still wide open for general groups.
In 2019, Ji, Natarajan, Vidick, Wright and Yuen resolved CEP in the negative. Quite remarkably, their result is deduced from complexity theory, and specifically from undecidability in certain quantum interactive proof systems. Inspired by their work, we suggest a novel interactive proof system which is related to the Aldous-Lyons conjecture in the following way: If the Aldous-Lyons conjecture was true, then every language in this interactive proof system is decidable. A key concept we introduce for this purpose is that of a Subgroup Test, which is our analogue of a Non-local Game. By providing a reduction from the Halting Problem to this new proof system, we refute the Aldous-Lyons conjecture.
These talks are based on joint work with Lewis Bowen, Alex Lubotzky, and Thomas Vidick.
Srivatsav Kunnawalkam Elayavalli, Strict comparison is the bringer of joy
Abstract: I will discuss recent joint work with co-authors Amrutam, Gao and Patchell, wherein we resolve the strict comparison problem for all acylindrically hypergolic groups with rapid decay, and discuss various applications to C*-algebras.
David Fisher, Lecture 1: Finiteness of totally geodesic submanifolds.
Abstract: I will give a survey of recent results on finiteness for totally geodesic submanifolds in various contexts with an emphasis on open questions. This is based on joint works in various combinations with Bader, Filip, Lafont, Lowe, Miller and Stover.
Lecture 2: Commensurators and connections
Abstract: I will discuss a few problems that have surprising connections to commensurators of discrete subgroups. This is joint work with Brody, Mj, and van Limbeek.
Josh Frisch, Randomized Stopping Times, Bounded Harmonic Functions on Groups, and Topological Realizations
Bounded harmonic functions on groups (and their alter ego, the Poisson boundary)—bounded functions such that f(x) is the average of f(xa), where a is chosen from some probability measure—are objects of key importance in random walks, dynamics, and probability. Two core techniques for understanding these ideas, used since the beginning of the subject, have been stopping times (randomized times when you stop your random walk) and realizations (classification of bounded harmonic functions by looking at a space where the random walk almost surely converges). I will discuss some new results linking these two concepts. This is joint work with Kunal Chawla.
Alex Furman, Lyapunov spectrum for some dynamical systems via Boundary theory.
Abstract: Given an ergodic pmp system $(X,m,T)$ and an integrable map $F:X\to SL(d,\mathbf{R})$ the Multiplicative Ergodic Theorem of Oseledets describes the asymptotic behavior of the products $F_n(x)=F(T^{n-1}x)\cdots F(Tx)F(x)$ by the Lyapunov spectrum $\Lambda=(\lambda_1\ge\dots\ge \lambda_d)$ with $\sum \lambda_i=0$, and a certain measurable family of flags on $\mathbf{R}^d$.
In this talk I will describe a joint work with Uri Bader, where we define a class of systems for which we can prove \emph{simplicity} of the spectrum: $\lambda_1>\lambda_2>\dots>\lambda_d$, and its \emph{continuity} under certain perturbations. This class of systems covers many interesting examples.
The proofs use ideas of "Boundary theory" for groups, that appear in the recent proofs of super-rigidity of representations of lattices and cocycles.
Jorge Garza Vargas, A new approach to strong convergence
Abstract: Let G be a discrete group and let lambda be its left regular representation. We say that a sequence rho_N of finite dimensional unitary representations of G converges strongly to lambda if for any element x of the group algebra of G we have that, as N goes to infinity, ||rho_N(x)|| converges to ||lambda(x)||, where ||.|| denotes the operator norm.
The phenomenon of strong convergence is ubiquitous in different areas of mathematics and it has enabled several recent breakthroughs in random graphs, spetral geometry, differential geometry, and operator algebras.
In this talk I will present a new approach to proving strong convergence results, based on elementary approximation theory and real analysis, that in the last year has delivered several state of the art results in the field.
This is joint work with Chi-Fang Chen, Joel Tropp and Ramon van Handel.
Tsachik Gelander, Lecture 1: IRS and SRS
Abstract: I will discuss the effect of the theory of random subgroups on the study of discrete subgroups of Lie groups and in particular lattices and arithmetic groups. An important role is played by Invariant Random Subgroups (IRS) and as well as by Stationary ones (SRS).
Lecture 2: Margulis infinite injectivity radius conjecture, with and without property (T).
Abstract: The celebrated normal subgroup theorem of Margulis states that if M=\Gamma\G/K is a finite volume irreducible locally symmetric manifold of rank >1, then any normal cover (except the simply connected one) has finite volume, i.e. a normal cover is a finite cover. Margulis suggested that the normality can be replaced by the much weaker requirement that the injectivity radius is bounded:
Theorem (Margulis conjecture): Let M be a finite volume irreducible locally symmetric manifold of rank >1, and M' a cover of M with bounded injectivity radius. Then M' is a finite cover of M.
When G has property (T) this result was proved in a joint work with Mikolaj Frakzyk and in fact, we obtained much stronger results. In the lack of property (T) (e.g. G=SL(2,R)xSL(2,R)), the result is considerably more challenging since novel techniques are required to replace the spectral gap provided by property (T).
This general case was proved recently in a joint work with Uri Bader and Arie Levit.
Alexander Kechris, Orbit equivalence relations and the compact action realization problem
Abstract: The study of orbit equivalence relations induced by Borel actions of countable groups on Polish (separable completely metrizable) spaces, and their orbit spaces, has been a very active area of research for several decades in various fields of mathematics, including ergodic theory, operator algebras, geometric group theory, combinatorics, probability and descriptive set theory. Many results in this area have been obtained using ergodic (measure theoretic) methods. After giving a basic introduction to this theory, I will focus on a new direction of topological nature that deals with the problem of realizing orbit equivalence relations by continuous actions on compact metrizable spaces and in particular subshifts. This also leads to considering a natural universal space for such actions and equivalence relations via subshifts and originates the study in this space of various important classes, especially the hyperfinite ones, which are those induced by actions of the group of integers. This is joint work with Josh Frisch, Forte Shinko and Zoltan Vidnyánszky.
Arie Levit, On stability of amenable groups
Abstract: In this talk we will explore the notion of stability, focusing on discrete amenable groups. A group is called stable if every almost-homomorphism is close to an actual homomorphism. Using different targets leads to different flavors of stability. We will discuss the following two classes: almost-actions (i.e. maps into finite symmetric groups) and almost-unitary representations (i.e. maps into finite-dimensional unitary groups). For amenable groups, there are powerful criteria at hand to determine whether a group is stable in either of those two senses, using the key notions of invariant random subgroups and characters, respectively. In both cases, the question of stability reduces to a suitable finitary approximation problem. We will survey those ideas and some of the results in the field.
Confined subgroups and irreducible lattices
Abstract: A subgroup H < G is called confined if G admits a compact subset K such that every conjugate of H intersects K at some point other than identity. We prove that every confined subgroup of an irreducible lattice in a higher rank semisimple Lie group has finite index. Since any non-trivial normal subgroup is confined, our result extends the Margulis normal subgroup theorem. We do not rely on Kazhdan’s property (T), and instead obtain a spectral gap from the product structure. More generally, we show that any confined discrete subgroup of a higher rank semisimple Lie group satisfying a certain irreducibility condition is a lattice. This extends the recent work of Fraczyk and Gelander, removing the property (T) assumption on the ambient group. The talk is based on joint work with Uri Bader and Tsachik Gelander.
Kevin Schreve, Homological growth and right-angled Coxeter groups
Abstract: A conjecture of Singer predicts that the L^2-Betti numbers of the universal cover of a closed aspherical manifold vanish outside of the middle dimension. I will talk about some recent results around this conjecture for manifolds constructed using right-angled Coxeter groups. I will also talk about connections to homological growth with various coefficients. Joint work with Grigori Avramidi and Boris Okun.
Giulio Tiozzo, The Poisson-Furstenberg boundary for random walks on groups
Abstract: The Poisson(-Furstenberg) boundary is a measure-theoretic object attached to a group equipped with a probability measure, and is closely related to the notion of harmonic function on the group. In many cases, the group is also endowed with a topological boundary arising from its geometric structure, and a recurring research theme is to identify the Poisson boundary with the topological boundary. In this talk, we will introduce the theory of Poisson boundaries and discuss how the identification problem is related to geometric properties of the related random walks, such as geodesic tracking.
The Poisson-Furstenberg boundary of hyperbolic groups without moment conditions
Abstract: In this talk, we prove that the Poisson boundary of a random walk with finite entropy on a non-elementary hyperbolic group can be identified with its hyperbolic boundary, without assuming any moment condition on the measure. In this generality, this identification result is new even for free groups. We will also discuss extensions of this result to other groups with hyperbolic properties, as well as discrete subgroups of Lie groups.
Joint with K. Chawla, B. Forghani, and J. Frisch.
Robin Tucker-Drob, Measure Equivalence of Baumslag-Solitar Groups
Abstract: We show that all non-amenable non-unimodular Baumslag-Solitar groups BS(r,s), 2≤|r|<s, are measure equivalent to each other, thereby completing the measure equivalence classification of Baumslag-Solitar groups. Consequently, each BS(r,s) belongs to one of three measure equivalence classes according to whether it is amenable (|r|=1 or |s|=1), virtually isomorphic to F_n\times Z (2\leq |r|=|s|), or non-amenable and non-unimodular (2\leq |r|<s). This is joint work with Damien Gaboriau, Antoine Poulin, Anush Tserunyan, and Konrad Wrobel.
Andrew Marks, Hyperfiniteness of Borel graphs of slow intermediate growth
Abstract: A Borel equivalence relation on a standard Borel space is hyperfinite if it is the increasing union of Borel equivalence relations with finite classes. The hyperfinite Borel equivalence relations are the simplest nontrivial class of Borel equivalence relations, by the Glimm-Effros dichotomy of Harrington-Kechris-Louveau. Yet many questions about hyperfiniteness remain open. For example, it is an open problem of Weiss (1984) whether the orbit equivalence relation of a Borel action of a countable amenable group is hyperfinite.
Some researchers have hoped we can use soft tools from Borel graph combinatorics and metric geometry to attack this problem, rather than relying on a sophisticated understanding of the structure of Folner sets and their tilings which have been key to much partial progress on Weiss's question. Recently, Bernshteyn and Yu made a significant advance in this direction by showing that every graph of polynomial growth is hyperfinite. Their result parallels the 2002 theorem of Jackson-Kechris-Louveau that Borel actions of polynomial growth groups are hyperfinite. We extend Bernshteyn and Yu's result to show there is a constant 0 < c < 1 such that every graph of growth less than is hyperfinite. This is joint work with Jan Grebik, Vaclav Rozhon, and Forte Shinko.
Volodymyr Nekrashevych, Conformal dimension and group actions
Abstract: We will discuss Ahlfors-regular dimension of metric spaces and its application to self-similar groups and their limit spaces. In particular, we will see how it can be used to prove the Liouville property (and hence amenability) for iterated monodromy groups of complex rational functions. This is a joint work with T. Zheng and N. Matte Bon.
Alina Vdovina, C*-algebras coming from buildings and their K-theory.
Abstract: We consider cross-product algebras of continuous functions on the boundary of buildings with cocompact actions. The main tool is to view buildings as universal covers of certain CW-complexes. We will find the generators and relations of the cross-product algebras and compute their K-theory. We will show how our algebras related Vaughan Jones' Pythagorean algebras.