Date/Time
Date(s) - 14/01/2025
3:30 pm - 4:30 pm

Britton Lectures

Dr. Henry Schenck
R.K. Brown Chair of Mathematics, Auburn University
Research Area: Algebra

Title: From Linear Programming to the Lefschetz Theorem

Abstract: In linear programming, the simplex method seeks to optimize the value of a linear functional over a polytope P. In this talk, we focus on understanding the geometry of P; it turns out that tools from algebra provide serious leverage in attacking the problem. We’ll start by investigating the face vector (number of vertices, edges, etc.) of a convex polytope P, recalling Euler’s famous formula for polytopes in dimension three. Switching gears, 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 the solution (sufficiency: Lou Billera & Carl Lee, necessity: Richard Stanley) to a famous conjecture using this machinery, which involves 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.