Stochastic Topology and Topological Statistical Mechanics

随机拓扑学是对随机形状的研究，而拓扑统计力学是研究复杂形状如何随着某些参数(如压力或熵)的变化而变化。这项工作涉及几个数学学科的交叉，包括拓扑学、几何学和组合学。它在数学之外也有应用。随机拓扑学为新兴的拓扑数据分析领域提供了一个适定的零假设。拓扑统计力学致力于寻找物质不同阶段的拓扑基础。该项目更广泛的影响集中在博士生的研究培训上，并为他们在学术界、数据科学和工业领域的职业生涯做好准备。作为该项目的一部分，PI将研究随机单纯复合体的新模型，包括随机超树。随机二维超树的基本群是随机群的一个有趣的模型，本课题的目的之一是高概率地证明这个群具有Kazhdan性质(T)。这是一个开放的问题，这些随机超树是否有一个比例限制；这样的限制，将是高维的类似Aldous的连续统随机树。PI还将调查复杂的配置空间，如硬盘或正方形的配置空间。从物理学的角度来看，它们很有趣，因为它们代表了硬球体系统的能量图景。该项目的一个主要部分是识别与周围环境同构的微不足道、非微不足道或同构的区域，以及粒子数量趋于无穷大时Betti数的渐近增长率。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。

Stochastic topology is the study of random shapes, and topological statistical mechanics is the study of how complicated shapes change as some parameter, such as pressure or entropy, varies. This work involves the intersection of several mathematical disciplines, including topology, geometry, and combinatorics. It also has applications outside of mathematics. Stochastic topology provides a well-posed null hypothesis and for the emerging area of topological data analysis. Topological statistical mechanics endeavors to find topological underpinnings of different phases of matter. The broader impacts of this project are focused on research training of PhD students, and to prepare them for careers in academia, data science, and industry. As part of this project, the PI will investigate new models of random simplicial complex, including random hypertrees. The fundamental group of a random 2-dimensional hypertree is an interesting model of random group, and one of the goals of the project is to prove that this group has Kazhdan's property (T) with high probability. It is an open-ended question whether these random hypertrees have a scaling limit; such a limit which would be higher-dimensional analogue of Aldous's continuum random tree. The PI will also investigate complicated configuration spaces, such as configuration spaces of hard disks or squares. These are interesting from the point of view of physics, since they represent the energy landscape for a hard-spheres system. Identifying the regimes where homology is trivial, nontrivial, or isomorphic to the ambient to the configuration space of points is a major part of the project, as are the asymptotic rate of growth of the Betti numbers as the number of particles tends to infinity.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.