Project A6: Definable reducibility



Principal investigators

Participating scientists

Summary

The study of analytic structures under Borel reducibility has become a central focus of descriptive set theory over the last two decades. In particular, the descriptive set-theoretic study of analytic equivalence relations has emerged as a fundamental tool in calibrating the complexity of classification problems throughout mathematics, and has led to solutions of a number of long-standing open problems. Nevertheless, our knowledge of the Borel reducibility hierarchy remains quite limited, even when restricted to the class of Borel equivalence relations.

The aim of this project is to further our knowledge of Borel reducibility of several specific types of analytic structures which have recently garnered a significant amount of interest. In the case of countable Borel equivalence relations, our work will hinge on refining known ergodic-theoretic rigidity arguments, particularly for actions of linear algebraic groups, as well as generalizing a number of well-known results on costs of groups and equivalence relations. Model-theoretic arguments involving the existence of ample generics for Polish groups may also come into play here. In the case of general analytic equivalence relations, we will rely on a combination of Baire category arguments with new graph-theoretic dichotomy theorems. For this reason, it will also be essential to further investigate properties of analytic graphs. Ergodic theory could well play a pivotal role here, too.

In particular, we plan to focus our attention on the following topics:
  • Borel equivalence relations.
  • Analytic graphs.
  • Groups of Borel automorphisms.
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