Triangulations of Convex Polytopes

Duke University, Durham NC

Investigators

Abstract

The PI studies several problems about triangulations of (lattice) polytopes which are motivated by connections to other fields of mathematics: The reconstruction problem is concerned with the relation of a polytope and its graph, which is of interest for linear optimization. Algebraic applications are concerned with polytopes whose vertices have all integral coordinates. The existence problem of unimodular triangulations and its variations has implications to computational algebra, algebraic geometry, and physical string theory. The methods used are mainly discrete geometric or combinatorial, with some ideas borrowed from topology and differential geometry. Convex polytopes are fascinating objects that have intrigued not only mathematicians since the ancient Greeks. The Platonic solids --- the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron --- are the most prominent and most mystified examples. Since George Dantzig's simplex method, linear optimization has been a driving force behind polytope theory, and it certainly is one of the most successful applications of mathematics to the `real world'. In the mean time, polytope theory found many more applications within mathematics, and beyond. Besides linear and integer programming, there are optimization, CAD, visualization, and even theoretical physics, just to name a few.