Descriptive Set Theory and Computability

在数学中遇到的一个重要问题是用不变量对某种类型的数学对象进行完全分类。描述集合论领域为研究这些类型的分类问题和比较它们的相对困难提供了一个总体框架。PI提出研究描述集理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社区的联系，并与研究生和年轻研究人员接触。该项目将支持在加州大学洛杉矶分校的研究生的培训。PI提出的研究韦斯的问题的顺从性和超有限性使用工具从格罗莫夫的渐近维数。这种方法韦斯的问题已经大大扩展和简化以前的结果的问题，并澄清他们的证明。这项调查的应用拓扑动力学和算子代数。PI还建议研究经典的几何悖论，如巴拿赫-塔斯基悖论和塔斯基圆平方问题。可测量组合学的最新进展催生了新的定理，使用来自博雷尔图上的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智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.