Descriptive set theory and recursion theory

在整个数学中遇到的一个基本问题是通过不变量将某种类型的数学对象完全分类。描述集合论为研究这类分类问题并比较它们的相对难度提供了一个一般框架。该领域取得了显著的成功，证明了简单分类类型的障碍的存在，校准了各种数学领域中分类问题的难度，并在理解所有分类问题的空间结构方面取得了进展。本研究与遍历理论、概率论、算子代数有着密切的联系，利用基于Borel确定性和递归理论的新技术研究了可数Borel等价关系的分类困难。这些工具已经解决了该领域中的几个重要问题，并有望解决该学科中需要新方法的问题，例如超有限等价关系的并增加问题，以及图灵等价是否普适的问题。这项工作也与描述性图组合数学的研究密切相关。最近在这一领域的突破为研究可数Borel等价关系领域的问题提供了新的组合方法，这些联系也启发了新的组合研究。

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.