Topological Dynamics and Countable Combinatorics

Dynamics is the study of objects in motion. Here, an "object" is represented by a point in some mathematical structure, and "motion" is understood to mean a group of symmetries of this structure. By considering different mathematical structures, ideas from dynamics interact with several other areas of mathematics, in particular logic, group theory, and combinatorics.

In turn, ideas from these fields give rise to new phenomena in dynamics. This project will focus on ideas coming from "countable combinatorics," broadly construed. Combinatorics is the study of discrete structures, as opposed to continuous ones, while the emphasis on "countable" combinatorics means that these discrete structures will be infinite, but still small enough that we can write down a list, or enumeration, of their elements.

Much research in the past two decades has elucidated how pigeon-hole principles on countable structures can yield examples of groups with simple dynamics; here, the group of symmetries will be large, i.e. uncountable. On the other hand, attention has shifted recently to considering combinatorial arguments on countable groups themselves; these can be used to show that countable groups always have wild dynamics.

The PI has two primary goals for this project. The first is to continue to develop the dictionary back and forth between combinatorial principles and dynamical properties of countable groups and more general Polish groups. In the case of countable groups, the existence of certain patterns on the group can lead to the construction of subshifts with interesting behavior, such as doubly minimal shifts or shifts disjoint from another given dynamical system.

For more general Polish groups, Ramsey-like principles can be used to produce interesting spaces of ultrafilter-like objects on which the group can act. These dynamical systems will then have important universal properties, such as being universal for minimal flows, or being a universal "completion flow," a dynamical object previously defined and investigated by the PI.

The second is to develop techniques to prove new theorems in countable combinatorics. Of particular interest are theorems asserting that various countable first-order structures have finite big Ramsey degrees. While the properties of interest are about countable objects, the techniques used can deal with extremely large objects, such as spaces of ultrafilters or forcing posets.

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.