Dynamics and Operator Algebras

This project is concerned with two mathematical fields. One of these studies various algebras in the analytic setting, which were originally invented by von Neumann to study quantum mechanics in physics. The other field studies the long time behavior of systems under evolution, such as the status of the solar system after one billion years and, more generally, symmetries of systems. Although these two fields look very different, in the last several years some surprising connections between them have been discovered, especially about infinite symmetries that can be approximated by finite symmetries. These connections have led to applications to both fields. The principal investigator aims to deepen the connections between operator algebras and dynamical systems. This includes the connection between the torsion-type invariant involving the first field and entropy invariant from the second field, and the study of Sylvester rank functions motivated by earlier investigations of connections between these two fields. The principal investigator will also study an embedding problem in the second field using mean dimension and Bernoulli actions. This project will help us understand the geometry and dynamics of various spaces and algebras. The newly-founded theory of invariants for sofic group actions, including both entropy and mean dimension, is developing rapidly. This research will give us a better understanding of this theory. Operator algebras have been found to be a powerful tool in the study of algebraic actions of non-abelian groups, replacing the commutative algebra tool in the study of algebraic actions of abelian groups. The project on the connection between the torsion-type invariant and entropy already has applications to the first theory and the entropy theory of such actions. Discoveries made under this project would lead to more applications and strengthen our understanding of a variety of mathematical phenomena. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.