Applications of Combinatorics to Problems of Geometry

Abstract

This thesis concerns applications of combinatorics to problems of geometry. Combinatorics and geometry are almost antipodal in the world of mathematics. Through the lens of combinatorics, mathematics is discrete and finitary, and we study graphs, sequences, trees and the processes which cause them to grow and modify and subdivide. Through the lens of geometry however, mathematics is a wild and continuous beast, and we study shapes, curves, surfaces and the processes which cause them to evolve and change and break apart. The intersection of combinatorics and geometry is the study of discrete, finitary processes which govern continuous, dynamical behaviour. Here, we will focus on exploiting these processes to solve two problems in Euclidean geometry, and two problems in algebraic geometry. Euclidean geometry is the study of regions in Euclidean n-dimensional space R^n. With so few constraints placed on the objects under consideration, this area is the home to some of the most notorious outstanding problems in all of mathematics. I believe that the best combinatorics problems are ones that anyone can understand but resist many good attempts at them. The Euclidean problems in this thesis certainly have this flavour, and one of them in particular lies at the center of an extremely active area of research surrounding stability of geometric inequalities. Algebraic geometry is the study of regions cut out by polynomial equations in C^n. Such regions display certain rigidities that arbitrary surfaces do not have, but they still vary continuously in families by varying the underlying equations. There are many profound open problems in the field, such as the Hodge conjecture and the Jacobian conjecture. At the same time, there are very basic questions about computing invariants of algebraic varieties which are still open and have important enumerative consequences --- we will focus on this latter type of problem here. I hope that the breadth of topics covered, spanning equivariant intersection theory, quantum cohomology, matroid theory, polytopes (occasionally infinite) and their subdivisions, lattice point enumeration, and the representation theory of S_n with n in C, will interest both the casual, and the ardent reader.