Bernstein-Sato polynomials and Hodge ideals of Algebraic Singularities

In this project we plan to study the Bernstein-Sato polynomials and the Hodge ideals associated to algebraic singularities. The BernsteinSato polynomial of an algebraic singularity is a difficult invariant to study, which is related to many other invariants of the singularity. For this object, we plan to study its roots from the geometry of the singularity. In particular, for plane curve singularities we want to determine the subsets of the roots of the Bernstein-Sato polynomial that are topological invariants. For higher dimensional singularities, which have resolution process that is well-understood, we would like to understand which divisors from a resolution contribute to the roots of the Bernstein-Sato. In a second part of this project, we would like to study the a fairly new invariant of a singularity, the Hodge ideals, introduced by Mustata and Popa. For plane curve singularities, and more generally, for isolated singularities, we would like to describe and compute effectively the Hodge ideals. As a consequence we would like to study the relation of the Hodge ideals with the Bernstein-Sato polynomial, the multiplier ideals and jumping numbers, and the spectrum of the singularity