Dynamics, Topology, and Combinatorics

University Of Florida, Gainesville FL

Investigators

Abstract

Dynamics originated in physics to model movement of particles in a space in continuous or discrete time. Abstract topological dynamics extends the notion of time from the groups of real numbers and integers to arbitrary, possibly infinite-dimensional, topological groups, and restricts the space to be bounded and to contain all its limit points. Abstract topological dynamics encompasses methods from and applications into numerous areas of mathematics. This project mainly investigates its intimate relationship with combinatorics, in particular, Ramsey theory. Every topological group admits a natural representation as a group of symmetries of some mathematical structure. It turns out that wild dynamical behaviour of the group corresponds to chaotic behaviour of the corresponding structure, that is, lack of Ramsey phenomena, and tamer dynamical behaviour corresponds to a bounded number of patterns approximating the undelying structure. This relationship is a rich source of questions both on the level of general theory and concrete examples. The intent of the project is to address both hand in hand while isolating instances suitable for undergraduate and graduate research. To a great extent, dynamical behaviour of a topological group can be read from its minimal flows, in particular from the projectively largest one, the universal minimal flow. It has previously been studied by functional analytical means, whereas the PI developed a dual approach via ultrafilters, for groups of automorphisms of discrete structure, and more generally near ultrafilters for general topological groups. The corresponding Ramsey phenomena have a direct translation into the ultrafilter language as well and clarify the connection between dynamics and combinatorics. The goal is to continue studying universal minimal flows via (near) ultrafilters in general, such as what topological spaces can be the universal minimal flows, as well as concrete examples of groups and underlying structures that lead the direction of the theory, for example, whether groups of automorphisms of discrete structures exhibit all possible dynamical behaviour, or whether metric structure (allowing for small error) bring in new 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.