Talks will be given on zoom, if you have the password you can join here. Please send an email to one of the organizers to get the the password.

If you would like to give a talk, please send the title, abstract and related papers (if available) of your proposed talk to one of the organizers by email.

Organizers: Amir Mohammadi, Anthony Sanchez, Brandon Seward

Fall 2022 Speakers Past Speakers

- September 29:

- October 6: Andrei Alpeev (Euler International Mathematical Institute)

Title: Amenabilty and random orders

Abstract: An invariant random order is a shift-invariant measure on the space of all orders on a group. It is easy to show that on an amenable group, any invariant random order could be invariantly extended to an invariant random total order. Recently, Glaner, Lin and Meyerovitch showed that this is no longer true for SL_3(Z). I will explain, how starting from their construction, one can show that this order extension property does not hold for non-amenable groups, and discuss an analog of this result for measure preserving equivalence relations.

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- October 13: Konrad Wrobel (McGill University)

Title: Orbit equivalence and wreath products

Abstract: We prove various antirigidity and rigidity results around the orbit equivalence of wreath product actions. Let F be a nonabelian free group. In particular, we show that the wreath products A ≀ F and B ≀ F are orbit equivalent for any pair of nontrivial amenable groups A, B. This is joint work with Robin Tucker-Drob.

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- October 20: Florent Ygouf (Tel Aviv University)

Title: Horospherical measures in the moduli space of abelian differentials

Abstract: The classification of horocycle invariant measures on finite volume hyperbolic surfaces with negative curvature is known since the work of Furstenberg and Dani in the seventies: they are either the Haar measure or are supported on periodic orbits. This problem fits inside the more general problem of the classification of horospherical measures in finite volume homogenous spaces.

In this talk, I will explain how similar questions arise in the moduli space of abelian differentials (and more generally in any affine invariant manifolds) and will discuss a notion of horospherical measures in that context. I will then report on progress toward a classification of those horospherical measures and related topological results. This is a joint work with J. Smillie, P. Smillie and B. Weiss.

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- October 27: Elad Sayag (Tel Aviv University)

Title: Entropy, ultralimits and Poisson boundaries

Abstract: In many important actions of groups there are no invariant measures. For example: the action of a free group on its boundary and the action of any discrete infinite group on itself. The problem we will discuss in this talk is 'On a given action, how invariant measure can be? '. Our measuring of non-invariance will be based on entropy (f-divergence).

In the talk I will describe the solution of this problem for the Free group acting on its boundary and on itself. For doing so we will introduce the notion of ultra-limit of G-spaces, and give a new description of the Poisson-Furstenberg boundary of (G,k) as an ultra-limit of G action on itself, with 'Abel sum' measures. Another application will be that amenable groups possess KL-almost-invariant measures (KL stands for the Kullback-Leibler divergence). All relevant notions, including the notion of Poisson-Furstenberg boundary and the notion of Ultra-filters will be explained during the talk. This is a master thesis work under the supervision of Yehuda Shalom.

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- November 3: Nachi Avraham-Re'em (Hebrew University of Jerusalem)

Title: Symmetric Stable Processes Indexed by Amenable Groups - Ergodicity, Mixing and Spectral Representation

Abstract: Stationary symmetric \alpha-stable (S\alpha S) processes is an important class of stochastic processes, including Gaussian processes, Cauchy processes and Lévy processes. In an analogy to that the ergodicity of a Gaussian process is determined by its spectral measure, it was shown by Rosinski and Samorodnitsky that the ergodicity of a stationary S\alpha S process is characterized by its spectral representation. While this result is known when the process is indexed by \mathbb{Z} or \mathbb{R}, the classical techniques fail when it comes to processes indexed by non-Abelian groups and it was an open question whether the ergodicity of stationary S\alpha S processes indexed by amenable groups admits a similar characterization.

In this talk I will introduce the fundamentals of stable processes, the ergodic theory behind their spectral representation, and the key ideas of the characterization of the ergodicity for processes indexed by amenable groups. If time permits, I will explain how to use a recent construction of A. Danilenko in order to prove the existence of weakly-mixing but not strongly-mixing stable processes indexed by many groups (Abelian groups, Heisenberg group).

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- November 10: Rogelio Niño (National Autonomous University of Mexico, Morelia)

Title: Arithmetic Kontsevich-Zorich monodromies of origamis

Abstract: We present families of origamis of genus 3 that have arithmetic Kontsevich-Zorich monodromy in the sense of Sarnak. It is known this is true for origamis of genus 2, however the techniques for higher genera should be different. We present an outline of the proof for the existence of these families.

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- November 17: Jayadev Athreya (University of Washington)

Location: AP&M 6402 and on Zoom

Title: Variance bounds for geometric counting functions

Abstract: Inspired by work of Rogers in the classical geometry of numbers, we'll describe how to obtain variance bounds for classical geometric counting problems in the settings of translation surfaces and hyperbolic surfaces, and give some applications to understanding correlations between special trajectories on these types of surfaces. Parts of this will be joint work with Y. Cheung and H. Masur; S. Fairchild and H. Masur; and F. Arana-Herrera, and all of this has been inspired by joint work with G. Margulis.

- November 22 (Tuesday) at 4 PM: Ruixiang Zhang (University of California Berkeley)

(Joint with the Combinatorics Seminar)

Location: AP&M 5829

Title: A nonabelian Brunn-Minkowski inequality

Abstract: The celebrated Brunn-Minkowski inequality states that for compact subsets X and Y of R^d, m(X + Y)^(1/d) >= m(X)^(1/d) + m(Y)^(1/d) where m is the Lebesgue measure. We will introduce a conjecture generalizing this inequality to every locally compact group where the exponent is believed to be sharp. In a joint work with Yifan Jing and Chieu-Minh Tran, we prove this conjecture for a large class of groups (including e.g. all real linear algebraic groups). We also prove that the general conjecture will follow from the simple Lie group case. For those groups where we do not know the conjecture yet (one typical example being the universal covering of SL_2(R)), we also obtain partial results. In this talk I will discuss this inequality and explain important ingredients, old and new, in our proof.

- November 24: Thanksgiving Holiday

- November 29 (Tuesday) at 2 PM: Camille Horbez (CNRS, Laboratoire de Mathématiques d'Orsay)

(Joint with the Functional Analysis Seminar)

Location: AP&M 7321

Title: Measure equivalence rigidity among the Higman groups

Abstract: The Higman groups were introduced in 1951 (by Higman) as the first examples of infinite finitely presented groups with no nontrivial finite quotient. They have a simple presentation, with k >= 4 generators, where two consecutive generators (considered cyclically) generate a Baumslag-Solitar subgroup. Higman groups have received a lot of attention and remain mysterious in many ways. We study them from the viewoint of measured group theory, and prove that many of them are superrigid for measure equivalence (a notion introduced by Gromov as a measure-theoretic analogue of quasi-isometry). I will explain the motivation and context behind this theorem, some consequences, both geometric (e.g. regarding the automorphisms of their Cayley graphs) and for associated von Neumann algebras. I will also present some of the tools arising in the proof. This is joint work with Jingyin Huang.

- December 1:

Winter 2023 Speakers

- January 12: David Aulicino (City University of New York and Brooklyn College)

- January 19: Karl Winsor (Harvard University)

- January 26:

- February 2: Tina Torkaman (Harvard University)

- February 9:

- February 16: Or Landesberg (Yale University)

- February 23: Homin Lee (Northwestern University)

- March 2: Félix Lequen (Cergy-Pontoise University)

Title: Bourgain's construction of finitely supported measures with regular Furstenberg measure

Abstract: The possible asymptotic distributions of a random dynamical system are described by stationary measures, and in this talk we will be interested in the properties of these measures — in particular, whether they are absolutely continuous. First, I will quickly describe the case of Bernoulli convolutions, which can be seen as generalisations of the Cantor middle third set, and then the case of random iterations of matrices in SL(2, R) acting on the real projective line, where the stationary measure is unique under certain conditions, and is called the Furstenberg measure. It had been conjectured that the Furstenberg measure is always singular when the random walk has a finite support. There have been several counter-examples, and the aim of the talk will be to describe that of Bourgain, where the measure even has a very regular density. I will explain why the construction works for any simple Lie group, using the work of Boutonnet, Ioana, and Salehi Golsefidy on local spectral gaps in simple Lie groups.

- March 9: Zvi Shem-Tov (Hebrew University of Jerusalem)

- March 16: