D-Modules and Commutative Algebra

The focus of this project is in algebraic geometry, one of the most varied areas of mathematics, permeating many different branches of science such as robotics, cosmology, and computer encryption. While the origins of algebraic geometry can be traced to the works of Euclid and Pythagoras, the focus of algebraic geometry today is often on singularities, which are points that are unusual when compared to their neighbors.

For example, in a figure-eight, the point of crossing is the only one of its kind on the curve – a singularity. Singularities appear frequently in nature, as the tip of a funnel cloud, in a black hole, or in abrupt changes of physical states such as a ball bouncing off a wall. They typically indicate states in which a given physical system becomes anomalous or unstable.

This project involves training of graduate and undergraduate students in research and educational activities, thus connecting research with the scientific advancement of the next generation.

There are three parts to this project on D-modules. At the center of the first part are D-modules that arise from embeddings of singular varieties into manifolds. The larger goals here include studying Hodge theoretic filtrations on these modules and their interplay with intersection homology.

With a view towards a conjecture of Hellus, the project will also aim at establishing a certain type of vanishing result. The second piece begins with a particular type of D-module that arises in the presence of a group action on a variety. For the resulting equivariant D-modules, the PI developed in previous NSF-funded work a categorical paradigm to study them in the toric case.

Adapting these tools to the more general case of tautological systems studied by Yau et al. in mirror symmetry is at the core of this aspect of the project. To investigate their functorial presentations, goals include finding a description of the moduli space and the period module. A third type of invariant to be investigated is the Bernstein–Sato polynomial of a singularity, which is closely related to solution counts via the wide-open Monodromy Conjecture.

An unexpected approach to this conjecture is introduced, based on a translation – via Frobenius thresholds in finite characteristic – to a new feature of hypergeometric D-modules. In all these parts, the PI will be directing thesis projects and engage undergraduates in research.

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.