Title: Combinatorics and Commutative Algebra

Abstract: This talk will give an overview of the spectacular success of algebraic methods in studying problems in discrete geometry and combinatorics. First, we'll discuss the face vector (number of vertices, edges, etc.) of a convex polytope and recall Euler's famous formula for polytopes of dimension 3. Then, we'll discuss graded rings, focusing on polynomial rings and quotients. Associated with a simplicial polytope P (every face is "like" a triangle) is a graded ring called the Stanley-Reisner ring, which "remembers" everything about P and gives a beautiful algebra/combinatorics dictionary. I will sketch Stanley's solution to a famous conjecture using this machinery and touch on connections between P and objects from algebraic geometry (toric varieties). No prior knowledge of the terms above will be assumed or needed for the talk.