Descriptive Set Theory, Geometrical Paradoxes, and Computability

在数学中遇到的一个基本问题是用不变量对某种类型的数学对象进行完全分类。描述集合论给出了一个研究此类分类问题和比较其相对困难的一般框架。PI提出研究描述集合论及其与几何悖论和可计算性的联系。主要的研究领域是建设性的解决方案，几何悖论，和复杂的数学分类问题，通过透镜的definability.The PI建议研究几何悖论使用的进展描述图组合学，并工作分类的复杂性行动的可数群遍历理论和描述集理论。最近在这些领域取得的进展，特别是在理论的可定义流图和超有限性的行动顺从的群体，导致了进展的几何悖论，如建设性的解决方案塔斯基的圆平方问题。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Lebesgue测度作为n-sphere上的可加性旋转不变Borel概率测度的唯一性。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。

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.