Descriptive Set Theory and Computability

描述性集合论和可计算性

基本信息

批准号：
2348208
负责人：
Andrew Marks
金额：
$ 34.72万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2024
资助国家：
美国
起止时间：
2024-04-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348208&HistoricalAwards=false
关键词：
Descriptive Set Theory Computability

项目摘要

An important problem encountered throughout mathematics is to completely classify some type of mathematical object by invariants. The field of descriptive set theory gives a general framework for studying these types of classification problems and comparing their relative difficulties. The PI proposes research in descriptive set theory and its connections with other mathematical fields including computability, operator algebras, topological dynamics, and ergodic theory. The PI will continue facilitating connections with these mathematical communities, and engaging with graduate students and young researchers. The project will support the training of graduate students at UCLA.The PI proposes research on Weiss's question on amenability and hyperfiniteness using tools from Gromov's theory of asymptotic dimension. This approach to Weiss's question has already greatly extended and simplified prior results on the problem and clarified their proofs. This investigation has applications to topological dynamics and operator algebras. The PI also proposes research on classical geometrical paradoxes such as the Banach-Tarski paradox and Tarski's circle squaring problem. Recent advances in measurable combinatorics have led to new theorems giving measurable solutions to these problems using combinatorial techniques from the study of flows and matching problems on Borel graphs.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.

数学中遇到的一个重要问题是通过不变量对某种类型的数学对象进行完全分类。描述性集合论领域为研究这些类型的分类问题并比较它们的相对难度提供了一个总体框架。 PI 提议研究描述性集合论及其与其他数学领域的联系，包括可计算性、算子代数、拓扑动力学和遍历理论。 PI 将继续促进与这些数学界的联系，并与研究生和年轻研究人员进行接触。该项目将支持加州大学洛杉矶分校研究生的培训。 PI 建议使用格罗莫夫渐近维数理论中的工具来研究 Weiss 的顺应性和超有限性问题。这种解决韦斯问题的方法已经极大地扩展和简化了该问题的先前结果，并澄清了他们的证明。这项研究可应用于拓扑动力学和算子代数。 PI还提出了对经典几何悖论的研究，例如Banach-Tarski悖论和Tarski圆平方问题。可测量组合学的最新进展催生了新的定理，使用来自 Borel 图上流和匹配问题研究的组合技术，为这些问题提供了可测量的解决方案。该奖项反映了 NSF 的法定使命，并通过使用基金会的知识进行评估，被认为值得支持。优点和更广泛的影响审查标准。