GRAZP 2024 seminar at IHP

This weekly research seminar is a part of the trimester at IHP,  Group actions and Rigidity: Around Zimmer program. 


The seminar will take place in IHP.  Typically, the seminar will be held on Monday and/or Wednesday at 3 pm or 3:30 pm.  Please check below for the exact time of each talk.

Organizers: Homin Lee and Thang Nguyen

List of talks

Week of April 22


Time: 3 pm
Title: Finite Fourier transform.

Abstract:

On a cyclic group of prime order, the non-principal Dirichlet characters together with their Fourier transforms are functions that vanish at 0 and whose modulus is constant outside 0. Do there exist other functions with these properties? I will explain the history of this question and give a positive answer for non-safe primes. The proof relies on Floer homology.

Week of May 6


Time: 3 pm
Title: Spectral theory of Anosov actions Abstract:  I will review the construction of a notion of discrete joint spectrum for Anosov abelian actions, and its link to SRB measure. The method uses anistropic Sobolev spaces and the notion of Taylor joint spectrum. Joint work with Bonthonneau, Hilgert and Weich.

Week of May 13

Time: 2 pm (unusual time)

Title: Entropies at infinity and applications

      Abstract: I will give several definitions of entropy at infinity for a dynamical system on a noncompact topological space. I will (maybe) show that they coincide. When the entropy at infinity is strictly smaller than the topological entropy of the system, in the case of geodesic flows in negative curvature, we obtain numerous interesting applications. Works in common with S Gouezel and S Tapie.

Time: 3 pm


Title: Compactifying hyperbolic representations all at onceAbstract: Let $\Gamma$ be a finitely generated group. A hyperbolic representation of $\Gamma$ is a non- elementary representation of $\Gamma$ in some $O(1,n)$ where $n$ can possibly be infinite. Such representations will be considered up to conjugations in $O(1,n)$.

Using continuous deformations of $O(1,n)$ into $O(1,\infty)$, any representation gives rise to a «homothety class » of irreducible representations in $O(1,\infty)$. We will prove that the space of homothety classes of hyperbolic representations is actually compact. So this provides a uniform way to compactify spaces of representations such as the Teichmüller space of closed surfaces.


This is a joint work with Christopher-Lloyd Simon.

Week of May 20

Week of June 3


Time: 3:30 pm
Title: The Normal Subgroup Theorem for Ã_2 latticesAbstract: An Ã_2 lattice is a group acting properly and cocompactly on a building of type Ã_2. Examples include uniform lattices in PGL_3(K), where K is a local field. However there are also more mysterious examples, which are not naturally embedded in a nondiscrete locally compact group. I will discuss an extension of a celebrated theorem of Margulis: in such a group, every normal subgroup is either trivial or of finite index. While the proof follows the general strategy of Margulis, I will explain how we are led to introduce new tools specific to this case. This is a joint work with Uri Bader and Alex Furman.

Week of June 17


Time: 3:30 pm 
Title: Local rigidity of actions of isometries on compact real analytic Riemannian manifolds
Abstract: In this work in collaboration with Z. Zhao (Nice), we consider analytic perturbations of isometries of an analytic Riemannian manifold $M$. We prove that, under some conditions, a finitely presented group of such small enough perturbations is analytically conjugate on $M$ to the same group of isometry it is a perturbation of. Our result relies on a ``Diophantine-like" condition, relating the actions of the isometry group and the eigenvalues of the Laplace-Beltrami operator. Our result generalizes Arnold-Herman's theorem about diffeomorphisms of the circle that are small perturbations of rotations.

Week of June 24


Title: EQUIDISTRIBUTION OF PERIODIC TORI

Abstract: Bowen and Margulis independently proved in the 70s that closed geodesics on compact hyper- bolic surfaces equidistribute towards the measure of maximal entropy. From a homogeneous dynamics point of view, this measure is the quotient of the Haar measure on PSL(2,R) modulo some discrete cocompact sugroup.

In a joint work with Jialun Li, we investigate the higher rank setting of this problem by taking a higher rank Lie group (like SL(d,R) for d ≥ 3) and by studying the dynamical properties of geodesic flows in higher rank : the so-called Weyl chamber flows and their induced diagonal action. We obtain an equidistribution formula of periodic tori (instead of closed orbits of the geodesic flow).


Title: Rigidity of fibrewise Anosov diffeomorphisms on principal torus bundles

Abstract: A fibrewise Anosov diffeomorphism is a fibre-preserving diffeomorphism on a fibre bundle that preserves an invariant stable and unstable splitting along the fibres. We discuss a result showing that every fibrewise Anosov diffeomorphism on a principal torus bundle is topologically conjugate to a map that is linear in the fibres, by a conjugacy that fibres over the identity on the base and is homotopic to the identity.



Last updated: April 4, 2024