Recent progress on rigity properties of higher rank diagonalizable actions and applications

TL;DR

This paper surveys recent advances in the rigidity properties of higher rank diagonalizable actions, highlighting their applications in number theory, quantum ergodicity, and Diophantine approximation.

Contribution

It provides a comprehensive overview of results and conjectures in the field, emphasizing new applications and connections.

Findings

Enhanced understanding of distribution of integer points

Progress in quantum unique ergodicity conjectures

New insights into Diophantine approximation

Abstract

The rigidity propeties of higher rank diagonalizable actions is a major theme in homogenous dynamics, with origins in work of Cassels and Swinnerton-Dyer in the 1950s and Furstenberg. We survey both results and conjectures regarding such actions, with emphasize on the applications of these results towards understanding the distribution of integer points on varieties, quantum unique ergodicity, and Diophantine approximations.