CAREER: Uncountable topological dynamics

University Of Florida, Gainesville FL

Investigators

Abstract

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Dynamics has its origins in physics: it studies motion in space, typically in discrete or continuous time. Its abstractions find numerous applications throughout mathematics. For example, objects can be understood or classified by their groups of symmetries and their actions (such as crystals or systems of differential equations), or they can be enclosed into a space and studied via its motions in it (such as flow of water on Earth and in its atmosphere). This project is in the realm of abstract topological dynamics, which extends the notion of time to any topological group and restricts spaces to compact Hausdorff. Its applications lie mostly in combinatorics, number theory, and measure theory. A priori, there is no bound on the size of the groups or spaces, however the majority of results have countability restrictions on both. The purpose of this project is to relax the countability conditions and to lay the foundations of uncountable topological dynamics. There are two major motives to do so: first, generalizations often bring clarification and simplification of a theory by revealing its true core and requires novel ideas that lead to new discoveries. Second, the uncountable is unavoidable as soon as we accept existence of countable infinity and even very natural dynamical systems of the discrete group of integers have complexity beyond any notion of countability. The core objects of abstract topological dynamics are minimal dynamical systems that can be thought of as building blocks of dynamical systems. In particular, the universal minimal flow captures the complexity of minimal dynamics of a given group since it quotients onto every minimal dynamical system. It is known that for infinite locally compact, non-compact groups, the universal minimal flow is non-metrizable. Yet, even the universal minimal flow of the discrete group of integers is poorly understood, in contrast to an extensively developed theory of its metrizable minimal flows. The aim of this project is to fill this gap and to provide explicit descriptions of universal minimal flows of foundational groups such as the integers, the integer lattice, or the general linear group. More generally, the PI will continue recently initiated investigation of interactions between universal minimal flows and operations on groups (such as products, short exact sequences, or inverse limits). On the other side of the spectrum, remarkable discoveries from the verge of the century of connections between finite combinatorics (Ramsey theory) and topological dynamics showed that universal minimal flows of non-locally compact (called infinite dimensional) groups that are essentially countable can be metrizable or even trivial. The PI proved analogous results for infinite dimensional groups with the countability assumption removed and will build further upon these results. The ultimate goal is to find a unifying theory for these two seemingly orthogonal dynamical behaviors of locally compact and infinite dimensional groups in the joint framework of ultrafilter dynamics established in discrete dynamics and extended to topological dynamics by the PI. The project lies on the boundary of set theory, topology, model theory, combinatorics, and algebra and offers ample training opportunities for graduate, undergraduate, and even talented high school students. 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.