D-modules and invariants of singularities

Singularities are points where a geometric object degenerates in some way. They play an important role in the study of algebraic geometry. This project is concerned with the use of certain novel techniques to study invariants of singularities. This theory provides an algebraic approach to the study of linear partial differential equations, but it has found many applications in algebraic geometry and related subjects. The goal of this project is, on one hand, to further study some recently introduced classes of singularities and, on the other hand, to explore new connections with zeta functions and invariants. The proposal provides ample opportunity for the PI to collaborate with graduate students and post-docs. He will also write a research monograph on this subject. This project will involve several different research directions. First, the PI will expand the study of k-rational and k-Du Bois singularities beyond the case of hypersurfaces, using the recently defined version of minimal exponent for locally complete intersections. In a different direction, he will investigate D-module theoretic invariants of singularities modulo powers of a given prime, that refine the test ideals and F-pure thresholds from positive characteristic, and that might provide an arithmetic counterpart to characteristic zero invariants, such as Hodge ideals and minimal exponents. A third direction concerns exploring connections between the motivic zeta function and the minimal exponent on one side, and between the topological zeta function and D-modules on the other side (with possible applications to the Strong Monodromy Conjecture). 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.