Seminar

Christopher O'Neill: "Classifying numerical semigroups using polyhedral geometry"

Abstract: A numerical semigroup is a subset of the natural numbers that is closed under addition.  There is a family of polyhedral cones C_m, called Kunz cones, for which each numerical semigroup with smallest positive element m corresponds to an integer point in C_m.  It has been shown that if two numerical semigroups correspond to points in the same face of C_m, they share many important properties, such as the number of minimal generators and the Betti numbers of their defining toric ideals.  In this way, the faces of the Kunz cones naturally partition the set of all numerical semigroups into "cells" within which any two numerical semigroups have similar algebraic structure.  In this talk, we survey what is known about the face structure of Kunz cones, and how studying Kunz cones can inform the classification of numerical semigroups.

Aldo Conca: "Two bounds on Castelnuovo-Mumford regularity"

I will report on bounds on the Castelnuovo-Mumford regularity for ideals with polynomial parametrization (joint work with F.Cioffi) and for ideals associated with general subspace arrangements (joint work with M.Tsakiris).