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）的变化而变化的。这项工作涉及到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。随机拓扑学为拓扑数据分析的新兴领域提供了一个完备的零假设。拓扑统计力学致力于寻找物质不同相的拓扑基础。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、数据科学和工业界的职业生涯做好准备。作为该项目的一部分，PI将研究随机简单复合体的新模型，包括随机超树。随机二维超树的基本群是一个有趣的随机群模型，课题的目标之一是证明该群具有高概率的Kazhdan性质(T)。这些随机超树是否有缩放极限是一个开放式的问题；这样的一个极限将是奥尔德斯连续统随机树的高维模拟。PI还将研究复杂的构型空间，如硬盘或正方形的构型空间。从物理学的角度来看，这些都很有趣，因为它们代表了硬球系统的能量景观。识别同态是平凡的、非平凡的或与点的环境构型空间同构的区域是该项目的主要部分，随着粒子数量趋于无穷大，Betti数的渐近增长率也是该项目的主要部分。该奖项反映了美国国家科学基金会的法定使命，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，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.