Making sandwiches: a novel invariant in D-module theory (David Lieberman, HMC)

In the field of commutative algebra, the principal object of study is (unsurprisingly) commutative algebras. A somewhat unintuitive fact is that results about commutative algebras can be gleaned from an associated non-commutative algebra whose generators are very analytic in nature. This object is called the ring of differential operators, often denoted by D. In a sense gives an algebraic way of constructing the partial derivative.

An important result in the study of D-modules is Bernstein’s inequality, first proved by Joseph Bernstein in the 1970’s. The result gives a lower bound on the filtered dimension of a D-module, which a provide insights about modules of commutative algebras. The goal of this talk is to present some novel singular settings where this inequality holds. To do this, we will introduce an invariant called sandwich Bernstein-Sato polynomials. These are analogous to a well studied object called the Bernstein-Sato polynomial, which is a generalization of the power rule taught in undergraduate calculus courses. Using sandwich Bernstein-Sato polynomials, we will show that Bernstein’s inequality holds true for the differential operators of the coordinate ring of the Segre product of projective spaces.