Filip Jonsson Kling: Face numbers of Simplicial Complexes

Abstract.

Given a simplicial complex with n vertices which is homeomorphic to a sphere of dimension d, what is the maximum number of faces of each dimension that it can have? In this talk, we will give an answer to this purely combinatorial question using tools from commutative algebra. In particular, we will introduce a wonderful tool called the Stanley-Reisner ring and show how it enables some of this interplay between combinatorial and algebraic information. If time permits, we will say some words on a stronger result called the g-theorem and how it may inspire the study of Lefschetz properties for graded algebras.