Stochastic Topology and Topological Statistical Mechanics

随机拓扑和拓扑统计力学

• 批准号: 2005630
• 负责人: Matthew Kahle
• 金额: $ 23.94万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005630&HistoricalAwards=false
• 关键词: Stochastic; Topology; Topological; Statistical; Mechanics

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.

随机拓扑学研究的是随机形状，而拓扑统计力学研究的是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。 它也有数学之外的应用。随机拓扑提供了一个适定的零假设，并为新兴领域的拓扑数据分析。拓扑统计力学致力于寻找物质不同相态的拓扑基础。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将研究随机单纯复形的新模型，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，该项目的目标之一是证明这个群具有高概率的Kazhdan性质（T）。这是一个开放式的问题，这些随机超树是否有一个缩放限制;这样的限制，这将是高维模拟奥尔德斯的连续随机树。PI还将研究复杂的配置空间，如硬盘或正方形的配置空间。从物理学的角度来看，这些是有趣的，因为它们代表了硬球系统的能量景观。该项目的一个主要部分是确定同调是平凡的，非平凡的，或同构于周围的配置空间的点的制度，因为是渐近增长率的贝蒂数的粒子数趋于无穷大。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。