Stochastic Topology and Topological Statistical Mechanics

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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.