Research Topics for Math 890 Algebra-geometry-combinatorics Seminar

A (convex) polytope is the convex hull of finitely many points in Rd. Alternatively, a polytope can be defined as the bounded intersection of finitely many halfspaces, that is, a polytope is the (bounded) solution set of a system of linear equations and inequalities. (This latter fact is the main non-geometric reason why people are interested in polytopes—linear systems are everywhere.) The main focus of my research centers around the seemingly simple task of computing the volume of a polytope, that is, volP := ∫ P dx. (If dimP 6= d, we have to adjust this definition by computing this integral in the affine span of P.) Volume computation is hard. To illustrate this statement, let me give an example. The nth Birkhoff polytope is the set of all n× n doubly-stochastic matrices: