Descriptive Dynamics and Borel Combinatorics of Group Actions

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The fields of descriptive set theory, dynamics, and combinatorics are established areas of mathematics. In recent years there has been increasing interest in the relatively new field of descriptive dynamics in which tools from these areas combine to study problems of common interest to these subjects. A particular focus is to study the structure of certain natural mathematical objects which occur as definable equivalence relations on the reals. These objects greatly generalize the objects of traditional study in descriptive set theory, and also extend naturally the objects, dynamical systems, studied in dynamics. The focus of this project is to use recently developed, cross-disciplinary techniques to study some of the fundamental properties of these objects. There are several important open questions for which the new techniques seem promising, and with which we can benchmark progress. Further progress in this area will lead to a better understanding of these areas of mathematics and their connections.A major focus of the proposal concerns the combinatorics of countable Borel equivalence relations. These are the relations generated by the Borel actions of countable groups. A natural plan of study is to proceed by increasing complexity of the group generating the relation. Even for relatively simple groups, the combinatorics of their group actions is interesting. As an example, recent methods allow for a computation of both the continuous and Borel chromatic numbers of free Bernoulli shifts by finitely generated abelian groups. There have also been recent developments to extend the proposers' result that all actions of countable abelian groups are hyperfinite to countable nilpotent groups (Schneider and Seward) and to some solvable, non-nilpotent groups (Conley, Jackson, Marks, Seward, and Tucker-Drob). The proofs show an interesting interplay between the properties of the equivalence relations and the combinatorial structure of complete sections in the relation. The project seeks to explore this interplay and to extend the current methods and results to actions of more general groups.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.