Senior Honors Thesis: "Face Numbers of Polytopes Constructed from Trivalent Trees"

Abstract: A convex polytope is the convex hull of a set of finite points in d-dimensional space, where the convex hull of a set S is the smallest convex set containing S. Polytotpes are of interest in various different fields. For example, in optimization, linear programming seeks to find vertices or maxima and minima on the boundary of an n-dimensional polytope. A face of a polytope P is the intersection of a supporting hyperplane and the polytope P. The f-vector of a n-dimensional polytope is a vector containing (n+1) entries such that the i-th entry is the number of i-dimensional faces of the polytope. Given a trivalent tree T with n leaves, define a polytope P_T as the convex hull of a set of n(n−1)/2 points, each encoding the edges used in a path between two leaves. We will talk about some conjectures on the f-vectors of theses polytopes.

Host: Laura Escobar