Descriptive set theory and recursion theory

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A fundamental problem encountered throughout mathematics is to completely classify some type of mathematical object by invariants. Descriptive set theory gives a general framework for studying such classification problems and comparing their relative difficulties. The field has had remarkable success, proving the existence of barriers to having simple types of classifications, calibrating the difficulty of classification problems in a variety of fields of mathematics, and in understanding the structure of the space of all classification problems. This study has had particularly close connections with ergodic theory, probability, and operator algebras.This project studies the difficulty of classifying countable Borel equivalence relations using new techniques based on Borel determinacy and recursion theory. These tools have already resolved several important questions in the field, and are promising candidates for attacking problems in the subject which are known to require new methods, such as the increasing unions problem for hyperfinite equivalence relations, and the question of whether Turing equivalence is universal. This work is also closely connected to the study of descriptive graph combinatorics. Recent breakthroughs in this area have yielded new combinatorial methods for studying problems in the field of countable Borel equivalence relations, and these connections have also inspired new combinatorial investigations.