Classification and invariants for Borel equivalence relations

Borel 等价关系的分类和不变量

• 批准号: 2246746
• 负责人: William Woodin
• 金额: $ 16.31万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246746&HistoricalAwards=false
• 关键词: Classification; invariants; Borel; equivalence; relations

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.

通过数学的一个共同线程是将对象集合分类为某些等效概念的问题。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将是等效的。该项目的主要目的是进一步发展“鲍尔等效关系”的理论。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，并确定何时成功分类。该项目将扩展理论，开发新方法，并将其应用于研究数学中的各种分类问题。 PI将通过教学，指导阅读，建议和指导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，以确定给定的分类问题何时有可能进行某些分类。分类不变的分析将通过公理设置理论的各种技术来促进，包括套件理论的对称模型和连续体的基本特征。具体而言，PI将进一步发展集合理论的对称模型之间的关系，其中选择的公理失败了，而Borel等效关系“可根据可数结构进行分类”。这种关系将用于解决有关这种等价关系的几个问题，并进一步分析其结构，直到减少鲍尔和鲍尔同构。此外，将研究不超级分类的一般框架，除了“按可数结构进行分类”之外，还将研究。这些奖项允许对以前被认为是无法分类的分类问题进行分类的有意义的调查。该奖项反映了NSF的法定任务，并使用基金会的智力优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。