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Project

New bridges between dynamics and coarse geometry

The project is concerned with novel interactions between dynamical systems and coarse geometry, with a particular focus on entropy, dimension, and spectral gaps.

Dynamical systems study evolution of processes. Depending on a researcher’s interest, one can study topological, ergodic, or measurable dynamics. A classical invariant of a topological dynamical system is its entropy, which measures the amount of chaos in the system’s evolution. For dynamics on probability spaces, the rate of averaging is governed by the spectral gap.

Coarse geometry originates from the need to identify metric spaces (often finitely generated groups or Lie groups) that resemble each other when viewed from afar, for example the integers and the reals. This paradigm enabled proofs of the Novikov and Borel conjectures for manifolds whose fundamental groups satisfy certain “coarse” properties.

In April 2020, Geller and Misiurewicz connected these areas, by pioneering the new direction of coarse dynamics. The first part of the project is a detailed study of the coarse notion of entropy that they introduce.

The second part is devoted to notions of dimension of a dynamical system, which are intimately related to asymptotic dimension in coarse geometry and nuclear dimension in operator algebras.

The third part is concerned with spectral gaps, and my recent application of them to the construction of new families of famous super-expander graphs.
 

Date:1 Nov 2021 →  Today
Keywords:coarse entropy, amenability dimension for actions of finitely generated groups on compact spaces, expander graph, spectral gap
Disciplines:Group theory and generalisations, Topological groups, Lie groups, Dynamical systems and ergodic theory, Functional analysis, Geometry not elsewhere classified