Speaker:  Maxwell Hosler, University of Kentucky

Title:  Alcolved polytopes

Qualifying Exam

We will discuss work of Lam and Postnikov on alcoved polytopes and some recent progress which extends their work.
 

The type-A affine Coxeter arrangement divides real space into unit simplices, called alcoves. Convex unions of these alcoves are called alcoved polytopes. We examine three additional ways of triangulating a particular family of alcoved polytopes called hypersimplices. It is shown they are all, in fact, identical to the alcoved triangulation, and that the logic behind them generalizes to all alcoved polytopes. This innovation gives us multiple ways to express the structure of alcoved polytopes, as well as drawing connections to commutative algebra.