Soficity, Dynamics, and Operator Algebras

This project concerns two mathematical fields. One of them studies the shape of geometric objects such as spheres using a tool that was originally invented by John von Neumann to study quantum mechanics in physics. The other field studies the long-time behavior of systems under evolution (e.g., the state of the solar system one billion years in the future) and, more generally, symmetries of systems. Although these two fields might appear to be quite different, in the last several years some surprising connections between them have been discovered, especially about those "infinite" symmetries that can be approximated by finite symmetries. These connections have led to applications in both fields. The principal investigator plans to deepen and broaden such connections. The principal investigator seeks to deepen the recently-found connection between the theory of invariants for manifolds via group von Neumann algebras and the theory of dynamical systems by introducing various relative invariants. This includes the connection between von Neumann-Lueck rank in the former theory and mean topological dimension in the latter and the connection between torsion-type invariant in the first theory and entropy in the second. The project will shed new light on the geometry and dynamics of various spaces and algebras. The newly-created theory of invariants for sofic group actions, including both entropy and mean dimension, is developing rapidly. The project will provides a better understanding of this theory. It will also deepen the interplay between different fields of mathematics. Operator algebras have become a powerful tool in the study of algebraic actions of nonabelian groups, replacing the commutative algebra tool that is used in the study of algebraic actions of abelian groups. The parts of the project on the connection between mean dimension and von Neumann-Lueck rank and on the connection between entropy and torsion-type invariants have already found applications to both the mean dimension and entropy theory of such group actions. Discoveries made in this project will lead to more applications and will strengthen one's understanding of a variety of mathematical phenomena.