Sam Payne: Triangulations with vanishing local h-polynomials

Abstract

Twenty-five years ago, Stanley introduced local h-polynomials for subdivisions of simplices, proved that the coefficients are non-negative integers, and posed the problem of characterizing triangulations for which this invariant vanishes. The work I will present is motivated by potential applications in other areas of mathematics (local h-polynomials now appear prominently in both algebraic and arithmetic geometry, through relations to intersection cohomology) yet the statements and proofs are purely combinatorial. The main results resolve Stanley's question in dimension 2 and 3, and give some promising first steps in higher dimensions. Joint with Elijah Gunther, Andre Moura, Jason Schuchardt, and Alan Stapledon.