Mikolaj Fraczyk, Infinite co-volume locally symmetric spaces
Finite covolume locally symmetric spaces are well understood thanks to Margulis' arithmeticity theorem and plethora of work on hyperbolic manifolds. While there are many results on infinite volume hyperbolic manifolds, in higher rank we know very little (with notable exception of quotients by Anosov subgroups where many rank one phenomena still hold). I will explain my recent work with Tsachik Gelander in which we prove that any infinite covolume higher rank locally symmetric space (HRLLS) has unbounded infectivity radius. Along the way I will try to squeeze in as many facts on HRLSS'ses as I can.
Alex Furman, Quotients of Poisson boundaries, Entropy, and Spectral gap
Fix a countable group $\Gamma$ with a probability measure $\mu$ on it. We propose to study the POSet\ of all measurable $\Gamma$-equivariant quotients equipped with the Entropy function on it.
We will discuss new results showing richness of this structure in the the case of free groups and surface groups.
Our methods rely on random walks on Lie groups, and some ideas that revolve around spectral gaps in compact groups.
This is joint work with Samuel Dodds (UIC).
Sebastian Hurtado (Part 1), The space of almost periodic actions in the line
We give a brief introduction to left-orderable groups, and describe the space of almost periodic actions of a group on the line (due to Bertrand Deroin). This is a compact space with a one dimensional lamination parametrizing the left-orders of a group. We will show how this space can be obtained via the study of random walks of homeomorphisms on the line.
Sebastian Hurtado (Part 2), Actions of higher rank lattices in dimension one
In joint work with Bertrand Deroin we study the left-orderability of higher rank lattices such as SL_2(Z[sqrt(2)]) and show that the actions of these groups on the circle by homeomorphisms come from the standard action of SL_2(R) in the circle.
Alireza Salehi Golsefidy (Parts 1 and 2), Super-approximation
Super-approximation is about the study of spectral gap property of a random-walk on congruence quotients of finitely generated subgroups of GL(n,Q). We will highlight the old and the new on this topic starting with the seminal work of Bourgain and Gamburd. We will finish with new results of myself and Srivatsa Srinivas, where we answer a question of Lindenstrauss and Varju in affirmative and go beyond that.
Mehrdad Kalantar (Part 1), An introduction to group C*-algebras
We give a quick introduction to C*-algebras, with the focus on the C*-algebras generated by unitary representations of discrete groups. We discuss connections between structural properties of group C*-algebras with the properties of their underlying groups. This talk is aimed at graduate students with no strong background in operator algebras.
Mehrdad Kalantar (Part 2), On invariant subalgebras of group C*-algebras
Let $\pi$ be a unitary representation of a discrete group $G$, and let $C^*_\pi(G)$ denote the C*-algebra generated by the unitaries $\pi(g)$, $g\in G$. This talk is concerned with "$G$-invariant" C*-subalgebras of $C^*_\pi(G)$, namely the C*-subalgebras that are invariant under conjugation by the unitaries $\pi(g)$. These objects can be considered as "generalized normal subgroups" of $G$ since for a subgroup $H\le G$, $C^*_\pi(H)$ is $G$-invariant iff $H$ is normal in $G$.
Our main result is that if $\pi$ is a non-amenable unitary representation of an irreducible lattice $G$ in the product of higher rank simple Lie groups, then any invariant $C^*$-subalgebra of $C^*_\pi(\Gamma)$ that is the range of an equivariant conditional expectation, is the C*-subalgebra generated by a normal subgroup. This is joint work with Nikolaos Panagopoulos.
Amir Mohammadi (Parts 1 and 2), Dynamics on homogeneous spaces: a quantitative account
Rigidity phenomena in homogeneous spaces have been extensively studied over the past few decades with several striking results and applications. We will give an overview of activities pertaining to the quantitative aspect of the analysis in this context with an emphasis on recent developments.
Andrés Navas, Distorted diffeomorphisms
An element of a finitely-generated group is said to be distorted if the word-length of its powers grows sublinearly. An element of a general group is said to be distorted if it is distorted inside a finitely-generated subgroup. This notion was introduced by Gromov and is worth studying in many frameworks. In this talk I will be interested in diffeomorphisms groups.
Calegary and Freedman showed that many homeomorphisms are distorted, However, in general, C^1 diffeomorphisms are not, for instance due to the existence of hyperbolic fixed points. Studying similar phenomena in higher regularity turns out to be interesting in the context of elliptic dynamics. In particular, we may address the following question: Given r > s > 1, does there exist undistorted C^r diffeomorphisms that are distorted inside the group of C^s diffeomorphisms? After a general discussion, we will focus on the 1-dimensional case of this question for r=2 and s=1, for which we solve it in the affirmative via the introduction of a new invariant, namely the asymptotic variation.
Amos Nevo, How to gauge the denseness of dense orbits
In an ergodic action of a countable group, almost every orbit is dense. But how dense these orbits actually are ?
A fundamental problem that appears in many different contexts is
to develop gauges that describe the denseness of orbits. Two sample questions that naturally arise are
1) are dense orbits typically equally dense ?
2) when can the dense orbits be shown to be optimally dense ?
Studying the effective denseness of orbits is a first step towards the much more ambitious goal of analysing their discrepancy in the space, at arbitrarily small scales.
We will consider these problems in the context of lattice actions on homogeneous spaces. This set-up is motivated by the challenge of establishing Diophantine approximation properties of general points on homogeneous algebraic varieties.
Based on joint work with Alex Gorodnik, and on joint work with Anish Ghosh and Alex Gorodnik.
Piotr Nowak (Parts 1 and 2), Property (T) and automorphism groups of free groups
The main focus of this talk will be Kazhdan's property (T). In talk 1 I will discuss Ozawa's characterization of property (T) via sums of squares, the approach to proving property (T) by means of semi-definite programming and the proof of property (T) for Aut(F_5), the automorphism group of the free group on 5 generators. In talk 2 I will present the proof of property (T) for Aut(F_n) for n\ge 6 and some applications.
Jesse Peterson, Properly proximal groups and actions
Properly proximal groups were introduced recently by R. Boutonnet, A. Ioana and the speaker in order to put Ozawa's class of biexact groups into a more general setting where operator algebraic indecomposability results would still hold. Earlier this year C. Ding, S. Kunnawalkam Elayavalli and the speaker showed that the notion of proper proximality also fits naturally into the framework of ergodic actions and von Neumann algebras. In this talk I will survey some of these results, focusing on examples and connections to solidity and solid ergodicity.
Alan Reid, Strongly dense representations of surface groups
In a 2012 paper Breuillard, Green, Guralnick and Tao introduced the notion of a group being strongly dense: if F=R or C then a subgroup G of SL(n,F) is strongly dense if any free subgroup H<G is Zariski dense in SL(n,F). In this talk we will discuss when representations of surface groups have (and have not) strongly dense image.
Scott Schmieding, Symbolic systems and their automorphism groups
This talk will be an introduction to symbolic dynamical systems and their automorphism groups. We'll discuss subshifts (over the integers), their groups of automorphisms, and survey some of the active topics and questions in the area.
Martin Schneider, Concentration of invariant means
A topological group G is called extremely amenable if every continuous action of G on a non-void compact Hausdorff space admits a fixed point. While no non-trivial locally compact group can be extremely amenable, many interesting examples of "infinite-dimensional" transformation groups are known to have this property. Most of these manifestations of extreme amenability have been exhibited using either structural Ramsey theory, or concentration of measure. The talk will be focused on some recent progress in the latter approach: I will discuss a new concentration inequality for convolution products of invariant means, based on a suitable adaptation of Azuma's martingale inequality, and show how this result can be used to prove the existence of fixed points. This method provides new examples of extremely amenable topological groups, which arise from von Neumann's continuous geometries.
Brandon Seward (Parts 1 and 2), Rokhlin entropy and direct products with Bernoulli shifts
In the first talk I will give a brief introduction to entropy. In the second, I will sketch a proof of the following result: if a free action is ergodic but not strongly ergodic and it has positive Rokhlin entropy, then the Rokhlin entropy of any direct product of this action with a Bernoulli shift will be equal to the Rokhlin entropy of the original action plus the Shannon entropy of the base of the Bernoulli shift.
Robin Tucker-Drob (Part 1), An introduction to quasi-measure-preserving equivalence relations and end-selection
In this talk I will define the Radon-Nikodym cocycle associated to a quasi-measure-preserving equivalence relation and introduce the mass-transport principle in this setting. I will also discuss treed equivalence relations and measurable end-selection for amenable subrelations of treed equivalence relations.
Robin Tucker-Drob (Part 2), Amenable subrelations of treed equivalence relations and the Paddle-ball lemma
We give a comprehensive structural analysis of amenable subrelations of a treed quasi-measure-preserving equivalence relation. This allows us to extend results that were previously only known in the measure-preserving setting. For example, we show that every nowhere-smooth amenable subrelation is contained in a unique maximal amenable subrelation. The two main ingredients are an extension of a criterion of Carrière and Ghys for nonamenability, along with a new Ping-Pong-style argument we call the "Paddle-ball lemma" that we use to apply this criterion in our setting. This is joint work with Anush Tserunyan.
Amanda Wilkens, Isomorphisms of Poisson systems over locally compact groups
We will introduce the Poisson system, a measure-preserving dynamical system made up of a Poisson point process and a group action. Ornstein and Weiss proved Poisson systems over many amenable groups were isomorphic in their 1987 paper. We will define and consider Poisson systems over non-discrete, non-compact, locally compact Polish groups, and prove by construction all Poisson systems over such a group are isomorphic, producing examples of isomorphisms for non-amenable group actions. As a corollary, we will show Poisson systems and products of Poisson systems are isomorphic. Additionally, the constructed isomorphisms have a nice algorithmic property, which we will discuss.
Tianyi Zheng (Parts 1 and 2), A flexible family of amenable groups
In these two lectures we will explain a flexible construction of diagonal products. The explicit structure allows one to understand quantitatively various characteristics. In joint work with J. Brieussel, these examples provide groups with prescribed random walk speed, spectral profile, and L_p space embedding distortion. More recently this construction is used in the work of A. Escalier to build prescribed quantitative orbit equivalence with the integers.