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.

贯穿数学的一个共同线索是将一组对象分类到某种等价概念的问题。一个成功的分类这些对象将是一个简单的列表的属性，这是很容易观察到的，使两个对象具有相同的属性实际上将是等同的。该项目的一个中心目标是进一步发展“博雷尔等价关系”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、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通用技术，以确定何时某些分类不变量可能用于给定的分类问题。分类不变量的分析将通过来自公理集合论的各种技术来促进，包括集合论的对称模型和连续统的基本特征。具体来说，PI将进一步发展集合论的对称模型之间的关系，其中选择公理失败，和博雷尔等价关系是“可分类的可数结构”。这一关系将被用来解决有关这类等价关系的几个问题，并进一步分析它们的结构，直到Borel约化和Borel同态。此外，广义框架的分类不变量，超越“可数结构分类”，将进行研究。这些允许一个有意义的调查可能的分类不变量的分类问题，以前被认为是unclassified.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。