Descriptive Aspects of Group Actions and Dynamics

Groups are a mathematical formalization of the notion of symmetry, which can also be used to model a wide variety of other phenomena, including the passage of time and the homogeneity of space. The ubiquity of this concept is central to its connection with dynamics, the study of systems that evolve in time and space. These processes can be either deterministic, as in the movement of planets, or stochastic, as in the throwing of dice.

The main purpose of this proposal is to study the interplay between dynamics and groups (group actions and dynamics) using the methods of logic, with a particular emphasis on the tools of descriptive set theory. Descriptive set theory precisely studies how complicated it is to define certain subsets, equivalences, and related objects. In particular, it can show whether certain simple classifications of objects exist or are impossible.

The principal investigator will focus on understanding the complexity of random walks and their Poisson Boundaries, the possible symmetries and joinings of dynamical systems, and the theory of Borel equivalence relations. Poisson Boundaries capture the notion of asymptotic indistinguishability for random walks. They also classify the space of an important set of functions on groups (the bounded harmonic functions).

When the boundary is trivial, this implies that in the long term all random walks look similar. One goal of this project is to make progress in understanding when this triviality occurs. Joinings represent a broad notion of symmetry, more general than morphisms or factor maps, that topological dynamical systems can have.

Understanding the joinings of a topological dynamical system to itself provides a rich perspective on the various symmetries that the system can possess. Borel equivalence relations are a formal way to precisely describe whether classification problems are easy or difficult. This project aims to further impact their study, as well as to initiate the exploration of linear algebraic analogues of Borel equivalence relations. Graduate students will be involved in this project.

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.