- Aaron Brown (Northwestern): Stiffness and Rigidity for inhomogeneous random walks
- Abstract: Motivated by numerous results on homogeneous and affine random walks, I will discuss some initial results and work-in-progress studying properties of random walks on manifolds generated by diffeomorphisms. I’ll primarily focus on describing certain dynamical criteria (which are verifiable in examples) on volume-preserving random walks which imply stiffness of the action. I may also discuss some directions towards establishing rigidity of invariant measures for the actions.
This is based on ongoing joint work with A. Eskin, S. Filip, and F. Rodriguez Hertz.
- Ilya Khayutin (Northwestern): Equidistribution and Arithmetic of Periodic Torus Orbits
- Abstract: This course is an introduction to the subject of periodic orbits of diagonalizable groups (tori). These periodic orbits carry arithmetic information and they play a crucial role in the study of the arithmetic of elliptic curves. Our goal is to discuss in depth several proofs of Linnik’s theorem about equidistribution of CM points on the modular curve with a view towards recent developments. The intended audience is graduate students who are interested in homogeneous dynamics, no advanced prerequisites will be assumed in number theory.
- Arie Levit (Yale): Rigidity, stability, and invariant random subgroups
- Abstract: I plan to discuss the group-theoretical notion of invariant random subgroups, focusing on two scenarios. For rank-one Lie groups I will talk about geometric and spectral properties of stabilizers in probability measure preserving actions. For discrete amenable groups, such as the lamplighter group, I will investigate the link between invariant random subgroups to questions of soficity and stability.
Outline (preliminary):
- Basic definitions and generalities on invariant random subgroups, e.g. connection with pmp actions, examples for discrete groups and Lie groups.
- Quantitative weak uniform discreteness and application to spectral gap of lattices (preprint of Gelander-Levit-Margulis).
- Ergodic theorem for hyperbolic groups, IRS in rank one Lie groups, the critical exponent and spectrum of the Laplacian (Abert-Glasner-Virag, Gekhtman-Levit).
- IRS vs soficity and stability (Becker-Lubotzky-Thom).
- Ergodic theorem for amenable groups, and stability for lamplighter groups and BH. Neumann groups via IRS (Levit-Lubotzky).
- Jesse Peterson (Vanderbilt): Higher-rank rigidity and von Neumann algebras
- Abstract: Associated to each probability measure-preserving action of a group is a finite von Neumann algebra known as the group-measure space construction. In 1976 Connes proved the remarkable result that, under some mild ergodicity type conditions, countably infinite amenable groups give isomorphic group-measure space constructions, and hence remember little about the group beyond its amenability. Moving beyond amenable groups, Connes later conjectured that the setting of lattices in higher-rank semi-simple Lie groups should be in stark contrast, with much of the group and its action being encoded in the group-measure space construction. In my minicourse I will discuss Connes' rigidity conjectures in detail, and I will survey the progress that has been made on them in the past 15 years.
The Einstein Institute of Mathematics
