# Project A5: Descriptive set theory and model theory

## Principal investigators

## Participating scientists

- Fabiana Castiblanco
- Yong Cheng
- Dr. Sean D. Cox
- Dr. Antongiulio Fornasiero
- Dr. Itay Kaplan
- Philipp Luecke
- Dr. Daniel Palacin

## Summary

The overarching goal of this project is to explore the ever-expanding interface between descriptive set theory and model theory.The first of our two primary focal points concerns applications of the two fields to the study of groups of automorphisms, particularly in topological dynamics. Recent work in descriptive set theory has demonstrated that relatively simple tools can be used to explore algebraic properties of descriptive- and measure-theoretic full groups. Unfortunately, such an approach is rarely useful in the context of topological dynamics, where the underlying structures lack the required level of independence. Meanwhile, recent work in model theory has led to similar results for automorphism groups of sufficiently generic structures, via an appropriate notion of independence inspired by model-theoretic stability. By combining the two approaches, we hope to obtain new algebraic results for topological full groups of suitably generic homeomorphisms and topological group actions.

The second of our focal points concerns applications of descriptive set theory to model theory. Recent work has successfully separated different notions of strong type by considering definable cardinals associated with the corresponding equivalence relations. Still more recently, these arguments have been simplified, generalized, and strengthened so as to give new applications to (model-theoretically) definable groups. We hope to continue to refine this work, in particular establishing dichotomy theorems for definable cardinals in terms of suitable algebraic criteria.

Finally, we also plan to extend previous work on independence in simple and NTP

_{2}-theories in order to get a better understanding of forking in such theories.