Combinatorics Seminar: From the Upper Bound Conjecture to Gorenstein linkage

Abstract: In 1957, Motzkin conjectured that the maximum number of faces possible for a polytope on n vertices in d-space is achieved by the convex hull of n points on the moment curve in d-space.  This conjecture, called the Upper Bound Conjecture, was proved by McMullen in 1970 and generalized by Stanley in 1975.  On the road to Stanley's proof, a correspondence between squarefree monomial ideals and simplicial complexes was born.  That correspondence is called the Stanley--Reisner correspondence.  It has come to occupy a central place in combinatorial algebraic geometry.  

A large share of this talk will be devoted to exposition on the Stanley--Reisner correspondence, Hilbert functions, and Cohen--Macaulayness and how they served the combinatorial aim of proving the Upper Bound Conjecture.  We will then define at most as many of the notions from Gorenstein linkage as we have to to state an open problem in that field.  We will reexpress an important special case of this problem in terms of questions about triangulations of polytopes,  and we will state some partial results from ongoing joint work with Jenna Rajchgot (McMaster) and Matt Satriano (Waterloo).

Host: Laura Escobar Vega