Descriptive set theory and recursion theory

在数学中遇到的一个基本问题是用不变量对某种类型的数学对象进行完全分类。描述集合论给出了一个研究此类分类问题和比较其相对困难的一般框架。该领域取得了显着的成功，证明了简单类型的分类存在障碍，校准了各种数学领域分类问题的难度，并理解了所有分类问题的空间结构。这个研究与遍历理论、概率论和算子代数有着特别密切的联系。这个项目研究了基于Borel确定性和递归理论的新技术对可数Borel等价关系进行分类的困难。这些工具已经解决了该领域的几个重要问题，并且是攻击该主题中已知需要新方法的问题的有希望的候选人，例如超有限等价关系的增加工会问题，以及图灵等价是否是普遍的问题。这项工作也密切相关的研究描述图组合。最近在这一领域的突破已经产生了新的组合方法研究问题领域的可数博雷尔等价关系，这些连接也激发了新的组合调查。

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.