Analytic Langlands Correspondence

This is a project in the field of algebraic geometry with connections to number theory and string theory. Algebraic geometry is the study of geometric objects defined by polynomial equations, and related mathematical structures. Three research projects will be undertaken. In the main project the PI will provide a generalization of the theory of automorphic forms, which is an important classical area with roots in number theory. This project provides research training opportunities for graduate students. In more detail, the main project will contribute to the analytic Langlands correspondence for curves over local fields. The goal is to study the action of Hecke operators on a space of Schwartz densities associated with the moduli stack of bundles on curves over local fields, and to relate the associated eigenfunctions and eigenvalues to objects equipped with an action of the corresponding Galois group. As part of this project, the PI will prove results on the behavior of Schwartz densities on the stack of bundles near points corresponding to stable and very stable bundles. A second project is related to the geometry of stable supercurves. The PI will develop a rigorous foundation for integrating the superstring supermeasure of the moduli space of supercurves. The third project is motivated by the homological mirror symmetry for symmetric powers of punctured spheres: the PI will construct the actions of various mapping class groups on categories associated with toric resolutions of certain toric hypersurface singularities and will find a relation of this picture to Ozsvath-Szabo's categorical knot invariants. 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.