Classification and invariants for Borel equivalence relations

数学中一个常见的问题是根据某种等价概念对一组对象进行分类。对这些对象的成功分类将是一个简单的属性列表，这些属性易于观察，因此具有相同属性的两个对象实际上是等效的。本课题的中心目标是进一步发展“Borel等价关系”理论。这是一个研究领域，它提供了一个严格的框架来分析数学中各种分类问题的复杂性，并确定何时可能或不可能成功分类。本项目将扩展理论，开发新的方法，并将其应用于研究数学中的各种分类问题。PI将通过教学、导读、咨询和指导等方式与本科生和研究生合作。PI还将参与组织会议和研讨会。该项目将开发和扩展一般技术，以确定何时某些分类不变量对于给定的分类问题是可能的。分类不变量的分析将由来自公理化集合论的各种技术促进，包括集合论的对称模型和连续体的基本特征。具体来说，PI将进一步发展集合论的对称模型（其中选择公理失效）与Borel等价关系（“可由可数结构分类”）之间的关系。这一关系将用于解决这类等价关系的若干问题，并进一步分析它们的结构直至Borel约简和Borel同态。此外，除了“可数结构分类”之外，还将研究分类不变量的广义框架。这些允许对以前被认为不可分类的分类问题的可能分类不变量进行有意义的调查。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

A common thread through mathematics is the problem of classifying a collection of objects up to some notion of equivalence. A successful classification of these objects would be a simple list of properties, which are easy to observe, so that two objects which have the same properties will in fact be equivalent. A central aim of this project is to further develop the theory of "Borel equivalence relations". This is a field of study which provides a rigorous framework to analyze the complexity of various classification problems in mathematics and to determine when a successful classification is possible or not. This project will expand the theory, develop new methods, and apply these to study various classification problems in mathematics. The PI will work with undergraduate and graduate students, through teaching, directed reading, advising, and mentoring. The PI will also be involved in organizing conferences and seminars.This project will develop and expand general techniques to determine when certain classifying invariants are possible for a given classification problem. The analysis of classifying invariants will be facilitated by various techniques coming from axiomatic set theory, including symmetric models of set theory and cardinal characteristics of the continuum. Specifically, the PI will further develop the relationship between symmetric models of set theory, in which the axiom of choice fails, and Borel equivalence relations which are "classifiable by countable structures". This relationship will be applied to settle several problems about such equivalence relations and to further analyze their structure up to Borel reductions and Borel homomorphisms. Furthermore, generalized frameworks of classifying invariants, beyond "classification by countable structures", will be studied. These allow for a meaningful investigation of the possible classifying invariants for classification problems which were previously considered unclassifiable.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.