Hema Srinivasan (University of Missouri)
Title: Sally type Numerical Semigroup ringsAbstract: A numerical semigroup of multiplicity e and width e − 1 is a Sally type numerical semigroup and the corresponding ring is the Sally type numerical semigroup ring. The name is inspired by July Sally’s paper on ”Good embedding for Gorenstein Rings”, where a good embedding is one where the associated graded ring is guaranteed to be Cohen Macaualay. She finishes this problem with a proof ending in a numerical semigroup of embedding dimension e−2 and width e − 1. In this talk, we will take up such a string of numerical semigroups and discuss all such semigroups of embedding dimension e − 2 as well as a class of these semigroups of smaller embedding dimension that are Gorenstein and construct their minimal resolutions. For the embedding dimension e − 2, we have some conjectures for the structure of Betti numbers of these semigroup rings. The first part of the talk is based on a WICA project with Dubey, Goel, Sahin and Singh. July Sally showed that a semigroup minimally generated by e, e + 1, e + 3, . . . , 2e − 1 is Gorenstein. This is a Sally type semigroup with exactly two consecutive numbers missing. We will show that for any k, there is exactly one of the Sally type semigroups of multiplicity e with exactly k consecutive numbers missing is Gorenstein and construct the minimal resolution of the semigroup ring. Their generators and resolution look very similar to the ones defined by arithmetic sequence although they are clearly not defined by an arithmetic sequence.
