This project is centers around understanding Riemannian manifolds via submanifolds. This topic and these methods have been central in Riemannian geometry since its conception, and have direct ties to problems and methods in geometric group theory. With regard to submanifolds, the PI seeks to further investigate the geometric data encoded by the primitive totally geodesic submanifolds of a fixed Riemannian n-manifold. The simplest case of 1-dimensional totally geodesic submanifolds is nothing more than the geodesic length spectrum and has garnered interest for more than 50 years. In addition, the geodesic length spectrum has strong connections with the eigenvalue spectrum of the Laplace-Beltrami operator and analysis on manifolds. In addition to this study, the PI plans on investigating which manifolds arise as totally geodesic submanifolds of a fixed Riemannian manifold. Two cases are locally symmetric manifolds of non-compact type and the moduli space of genus g curves with n marked points. In the latter case, most prominent are the Teichmuller curves and their associated fundamental groups Veech groups, which are interesting geometrically, algebro-geometrically, and dynamically. In addition, the PI seeks obstructions or special properties of submanifolds of a fixed manifold, not only in an effort to understand the fixed manifold but also to better understand obstructions to isometrically immersing manifolds into other manifolds. Lastly, the PI plans to investigate the asymptotic behavior of geometric counting functions, what geometric data is encoded by these functions, and ties to this topic outside of geometry. These function counting the totally geodesic manifolds of a fixed type as a function of volume and a basic entities, which in the simplest cases are known to have ties to geometric dynamics and applications to number theory.
The objects that arise in the present proposal permeate mathematics as fundamental examples tied to basic problems and areas. They were fathered not by mathematics but from pure and applied science. Moreover, over time, have been proven to be centrally important not only in mathematics but in physics, chemistry, and computer science. The PI hopes to foster further these important connections not only directly with specific results but also in disseminating the core ideaology of the PI's proposal.