Research

Research Interests

• F-singularities

When studying the singularities of surfaces over fields with positive characteristic, analytical notions of smoothness are not applicable. However, there are invariants associated to the ideals defining these surfaces that are of great interest, one of the most prominent being the F-pure threshold. Despite being studied rigorously for decades, they are still difficult to compute. 

• Can we implement elementary algorithms to compute F-pure thresholds of certain polynomials?

• Can we compute the F-pure threshold of more general classes of polynomials?

• D-modules

The algebraic properties of rings of differential operators was first investigated by Bernstein, to show the existence of solutions to the Bernstein functional equation over a polynomial ring. These solutions consist of a differential operator, and a polynomial, and the polynomial solution of least degree is called the Bernstein-Sato polynomial.

• • Can we find sufficient conditions on a ring (finitely generated k-algebra) to confirm the existence of solutions to the Bernstein functional equation?

  • Can we compute the Bernstein-Sato polynomial? 

  • What can we say about the structure of a ring R as a module over its ring of differential operators D?