Descriptive Set Theory, Geometrical Paradoxes, and Computability

在整个数学中遇到的一个基本问题是通过不变量将某种类型的数学对象完全分类。描述集合论为研究这类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几何悖论和可计算性的联系。主要研究领域是几何悖论的构造性解决方案，以及从可定义性的角度看待数学分类问题的复杂性。PI建议利用描述图组合学中的进展来研究几何悖论，并在遍历理论和描述集合论中对可数群的动作的复杂性进行分类。这些领域的最新进展，特别是在图中可定义流理论和顺从群的行动的超有限性方面的进展，导致了几何悖论的进展，例如塔尔斯基圆方问题的建设性解。PI建议在这项工作的基础上进一步研究这方面的问题，例如勒贝格度量作为n球面上的有限可加旋转不变Borel概率度量的唯一性。这一奖项反映了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。

A fundamental problem encountered throughout mathematics is to completely classify some type of mathematical object by invariants. Descriptive set theory gives a general framework for studying such classification problems and comparing their relative difficulties. The PI proposes research in descriptive set theory and its connections with geometrical paradoxes and computability. The main areas of study are constructive solutions to geometrical paradoxes, and the complexity of classification problems in mathematics viewed through the lens of definability.The PI proposes studying geometrical paradoxes using advances made in descriptive graph combinatorics, and work classifying the complexity of actions of countable groups in ergodic theory and descriptive set theory. Recent progress in these fields particularly in the theory of definable flows in graphs and on the hyperfiniteness of actions of amenable groups, has led to progress in geometrical paradoxes such as a constructive solution to Tarski's circle squaring problem. The PI proposes building on this work to study further problems in this vein such as the uniqueness of Lebesgue measure as a finitely additive rotation invariant Borel probability measure on the n-sphere.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.