Abstract

Speaker: Hal Schenck, Texas A&M University

Title: Combinatorics and commutative algebra

Abstract:

In this expository talk, I'll give an introduction to the use of algebraic methods in discrete geometry and combinatorics. We'll begin with the f (face) vector of a simplicial polytope P; as you might expect the f-vector counts the number of faces of each dimension. Example: f(octahedron) = (6,12,8)= 6 vertices, 12 edges, 8 triangles. The motivating conjecture is due to Motzkin, from the 1950's: Does there exist a (pointwise) biggest f-vector, if we fix the number of vertices and dimension of P.
Having set up the problem, we translate (following Stanley) into commutative algebra. So I'll discuss graded rings (specifically, polynomial rings and quotients), free resolutions, and Hilbert series. Associated to a simplicial complex is a graded ring (the Stanley-Reisner ring) whose algebra reflects combinatorial and topological aspects of the complex, and which gives us a (really beautiful) algebra/combinatorics dictionary. The punchline of the talk will be an answer (due to Stanley) to Motzkins question.

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