# Global rigidity of actions by higher-rank groups

May 16 to May 20, 2022

at the

American Institute of Mathematics, San Jose, California

organized by

Aaron Brown, David Fisher, Ralf Spatzier, and Zhiren Wang

## Original Announcement

This workshop will be devoted to actions of higher rank groups such as $SL(n, Z)$, $n \geq 3$ and $Z^d$, $d \geq 2$. The general theme is the global rigidity or classification of such actions (at times satisfying additional dynamical hypotheses) up to smooth changes of coordinates. Two major motivations in this area are the Zimmer program and the Katok-Spatzier conjecture, which respectively concern the classifications of actions by lattices in higher rank Lie groups and Anosov actions by higher rank abelian groups. During the last few years, there have been numerous breakthroughs for both type of groups, including the proof of Zimmer's conjecture for $SL(n, Z)$ and cocompact lattices of higher rank $R$-split simple groups and recent work advancing the classification of abelian Anosov actions. A large volume of new techniques have appeared in various directions surrounding these programs, including functional analysis on groups, homogeneous dynamics, smooth ergodic theory, and invariant algebraic or geometric structures. Given these developments, we expect future progress on various global rigidity conjectures. The goals of the workshop will include:
• Presentations on current state of the art techniques to build invariant algebraic/geometric structures;
• Construction, classification, and investigation of the properties of exotic actions;
• Exchange of techniques developed by different research groups with the goal of developing new collaborations and making further progress in global rigidity programs.

## Material from the workshop

A list of participants.

The workshop schedule.

A list of open problems.