RII Track-4: Applied Symplectic Topology

University Of Mississippi, University MS

Investigators

Abstract

Topology is the branch of mathematics that, broadly speaking, studies the shape of space. This notion of "shape" can be very abstract: symplectic topology is the study of the "shape" of energy surfaces of physical systems with conservation of energy. An important question in statistics is to determine the "shape" of data. Classical tools include, for instance, linear regression to find the line of best fit -- but what if the data doesn't have such a simple shape? Recent developments have brought powerful tools and ideas from abstract topology to bear on problems in statistics. This fellowship will develop a new collaboration between the University of Mississippi and the TGDA@OSU interdepartmental research group on applied topology at the Ohio State University in order to adapt these methods to problems in symplectic topology, thus opening up a new field of study: probabilistic symplectic topology. The methods developed will be among the first probabilistic tools used in symplectic topology and will open a new vista of questions to consider. This project will expand the PI's ability to collaborate with colleagues in engineering and the sciences, strengthen the UM's research program in topology and dynamics, and enhance graduate and undergraduate education. The goal of this project is to combine the techniques and ideas of applied topology with problems of symplectic topology. One of the key problems in symplectic topology concerns the existence of periodic orbits of a Hamiltonian system within a given range of actions and indices, or to be more precise, from filtered chain complexes. From this are constructed a large number of symplectic invariants such as symplectic capacities. The PI will investigate the potential to construct and compute new symplectic invariants using the tools of applied topology, such as persistent homology. The PI will also study the distribution of symplectic capacities for a (suitably defined) random convex domain in 4-space, both numerically by using the computational methods of applied topology, and theoretically, using the techniques developed by Kahle to study random complexes. This will open up many questions in the new area of probabilistic symplectic topology. Furthermore, a development of the framework of discretized symplectic topology will enable the effective computation of invariants, allowing for applications in the study of concrete dynamical systems. This project, through the PI and his graduate student's exposure to the techniques of applied topology, will additionally bring a new area of expertise to Mississippi. 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.