Month: April 2020

By awb1472 on Talks, Uncategorized

05/11: Sebastian Hurtado

Random walks by homeomorphisms on the line and left-orderability of lattices

Abstract: The standard random walk in the integers is known to be recurrent, it passes through every integer infinitely many times. We will discuss a generalization of this theorem for random walks given by random composition of homeomorphisms of the line (due to Deroin-Navas-Kleptsyn-Parwani) and some applications of this theorem to the theory of left–orderable groups. Our main result is that a lattice in a real semi-simple Lie group of higher rank and finite center is not a left–orderable group (equivalently, every action of such lattice in the line by homeomorphisms is trivial), a conjecture due to Witte-Morris and Ghys.  (Joint work with Bertrand Deroin).

 

Slides

By awb1472 on Talks

05/04: Nikos Frantzikinakis

Ergodic properties of bounded multiplicative functions and applications

Abstract: The Möbius and the Liouville function are multiplicative functions that encode important information related to distributional properties of the prime numbers. It is widely believed that their non-zero values fluctuate between plus and minus one in a random way, and two conjectures in this direction, one by Chowla and another by Sarnak, have attracted a lot of interest in recent years. In this talk, we shall see how one can make progress towards these conjectures by using ergodic theory in order to analyze feedback originating from analytic number theory. It turns out that key to the success of this method is an in depth understanding of structural properties of measure preserving systems naturally associated with bounded multiplicative functions. I will present what is known and what remains to be determined about such systems. The talk is based on joint work with Bernard Host.

Slides

By awb1472 on Talks

04/27: Zhiyuan Zhang

Exponential mixing of 3D Anosov flows

Abstract:  We show that a topologically mixing C^\infty Anosov flow on a 3 dimensional compact manifold is exponential mixing with respect to any equilibrium measure with Holder potential. This is a joint work with Masato Tsujii.

Slides

 

By awb1472 on Talks

04/20: Brian Chung

Stationary measure and orbit closure classification for random walks on surfaces

Abstract: We study the problem of classifying stationary measures and orbit closures for non-abelian action on surfaces. Using a result of Brown and Rodriguez Hertz, we show that under a certain average growth condition, the orbit closures are either finite or dense. Moreover, every infinite orbit equidistributes on the surface. This is analogous to the results of Benoist-Quint and Eskin-Lindenstrauss in the homogeneous setting, and the result of Eskin-Mirzakhani in the setting of moduli spaces of translation surfaces.

We then consider the problem of verifying this growth condition in concrete settings. In particular, we apply the theorem to two settings, namely discrete perturbations of the standard map and the Out(F2)-action on a certain character variety. We verify the growth condition analytically in the former setting, and verify numerically in the latter setting.

 

Slides

By awb1472 on Talks, Uncategorized

04/13: Cagri Sert

Extremal behaviour and spectral radius of random products of matrices

Abstract: I will survey some recent progress on random matrix products theory. In a first part, I will talk about some rigidity phenomenon for the laws of random products exhibiting an extremal behaviour (e.g. fastest possible norm growth). In a second part, I will focus on spectral radius of random products and mention two recent limit theorems: law of large numbers (without irreducibility condition), and for rank one groups, large deviation principle.

Based on joint works with Richard Aoun, Jairo Bochi, Emmanuel Breuillard and Alessandro Sisto.

Slides

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04/06: Arie Levit

Quantitative weak uniform discreteness

Abstract: I will discuss a quantitative variant of the classical Kazhdan-Margulis theorem generalized to probability measure preserving actions of semisimple groups over local fields. More precisely, the probability that the stabilizer of a random point admits a non-trivial intersection with a small r-neighborhood of the identity is at most b r^d, for some explicit constants b, d>0 which depend only on the semisimple group in question. Our proof involves some of the original ideas of Kazhdan and Margulis, combined with methods of the so-called Margulis functions as well as (c,\alpha)-good functions on varieties. As an application, we present a new unified proof of the fact that all lattices in these groups are weakly cocompact, i.e admit a spectral gap.

The talk is based on a recent preprint joint with T. Gelander and G.A. Margulis.