|Location:||MSRI: Simons Auditorium|
Given a finite subset A of positive integers, let S(A) denote the induced semigroup ring. In other words, S(A) is the coordinate ring of the monomial
curve parametrized by A. For a positive integer j, write A+(j) for the
subset obtain by adding j to each element in A. In this talk, we will discuss the conjecture that the Betti numbers of the semigroup ring S(A+(j)) are eventually periodic in j. In particular, this conjecture implies that the first Betti number of S(A+(j)), which is the minimal number of equations defining the associated monomial curve, is eventually periodic in j and hence bounded for all j.