Descriptive Set Theory and Computability

在整个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一般框架。PI建议研究描述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。国际数学联合会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。该项目将支持加州大学洛杉矶分校研究生的培养。PI建议使用Gromov的渐近维度理论中的工具来研究Weiss关于适应性和超有限性的问题。这种处理Weiss问题的方法已经大大扩展和简化了前人关于这一问题的结果，并澄清了它们的证明。这项研究在拓扑动力学和算子代数中都有应用。PI还提出了对经典几何悖论的研究，如Banach-Tarski悖论和Tarski圆平方问题。在Borel图上的流动和匹配问题的研究中，可测量组合学的最新进展导致了使用组合技术为这些问题提供可测量解决方案的新定理。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.