Abstracts

Week of November 12, 2023

November 13, 2023
Group Actions, Geometry and Dynamics Stationary probability measures on projective spaces 4:00pm -
KT801

We give a description of stationary probability measures on projective spaces for
an iid random walk on GLd(R) without any algebraic assumptions. This is done
in two parts. In a first part, we study the case (non-critical or block-dominated
case) where the random walk has distinct deterministic exponents in the sense of
Furstenberg-Kifer-Hennion. In a second part (critical case), we show that if the
random walk has only one deterministic exponent, then any stationary probability
measure on the projective space lives on a subspace on which the ambient group of
the random walk acts completely reducibly. This connects the critical setting with
the work of Guivarc’h-Raugi and Benoist-Quint. Combination of all these works
allow to get a description of stationary probability measures. Joint works with Richard Aoun.
November 15, 2023
Colloquium Geometry and Dynamics of Anosov groups 4:00pm -
KT 219
Francois Labourie introduced the theory of Anosov subgroups of semi-simple Lie groups in his seminal work 
on Hitchin representations. Anosov groups are now widely recognized as the higher rank generalization of convex cocompact 
subgroups of rank one Lie groups. (For our purposes, all higher rank Lie groups will be SL(d,R) and rank one Lie groups will 
all be PSL(2,R), PSL(2,C) or SO(d,1).) We will review classical results on convex cocompact subgroups of rank one Lie groups 
and discuss how they generalize into the setting of Anosov groups. 
November 16, 2023
Group Actions, Geometry and Dynamics Transverse groups and relatively Anosov groups 4:00pm -
KT205

Transverse subgroups of semi-simple Lie groups include all discrete subgroups of rank one Lie groups and Anosov and relatively Anosov subgroups of higher rank Lie groups. Every transverse group comes equipped with a  family of “geodesic flows’’ associated to linear functionals on the Cartan subspace. The assumption of transversality allows for the construction of well-behaved Patterson-Sullivan measures on the limit set and associated Bowen-Margulis-Sullivan measures on the flows.  We discuss a Hopf-Tsuji-Sullivan dichotomy in this context. Finally, we show that relatively Anosov groups live on the divergent side of this dichotomy (Joint work with Tengren Zhang and Andy Zimmer)
Analysis Generalized Bessel Functions in High Dimension 4:00pm -
KT 219

Many quantities of interest throughout mathematics are basically Bessel functions of a vector argument. As a result, natural asymptotic questions in various fields can be reduced to understanding the behavior of multivariable Bessel functions as the dimension of the domain grows large. In this talk, I’ll introduce the theory of generalized Bessel functions and describe how these functions form a useful link between subjects as diverse as random matrix theory, harmonic analysis and integrable systems. Then I’ll present a new result that characterizes the high-dimensional asymptotics of generalized Bessel functions by studying the hydrodynamics of associated stochastic processes. Joint work with Jiaoyang Huang (https://arxiv.org/abs/2305.04131).
Learning seminar on D-modules 4:00pm -
KT 801

This is the 9th lecture in the series.
November 17, 2023
Friday Morning Seminar Friday Morning Seminar 10:00am -
KT 801

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!
Geometric Analysis and Application Positive scalar curvature metric and aspherical summands 2:00pm -
KT 906

Abstract: It has been a classical question which manifolds admit Riemannian metrics with positive scalar curvature. A manifold is called aspherical if it has contractible universal cover. By works of Chodosh—Li, Gromov, and Chodosh—Li—Liokumovich, for n = 4, 5, the connected sum of a closed aspherical n-manifold with an arbitrary closed manifold does not admit a metric with positive scalar curvature. We prove that for n = 3,4,5, the connected sum of a closed aspherical n-manifold with an arbitrary non-compact manifold does not admit a complete metric with nonnegative scalar curvature. In particular, a special case of our result answers a question of Gromov. This is joint work with Jianchun Chu and Jintian Zhu.