Title: Lattice points, oriented matroids, and zonotopes
Advisor: Dr. Josephine Yu, School of Mathematics, Georgia Institute of Technology
Committee:
Dr. Matthew Baker, School of Mathematics, Georgia Institute of Technology
Dr. Galyna Livshyts, School of Mathematics, Georgia Institute of Technology
Dr. Santanu Dey, School of Industrial and Systems Engineering, Georgia Institute of Technology
Dr. Prasad Tetali, School of Mathematics, Georgia Institute of Technology
Reader: Dr. Francisco Santos, Departamento de Matematicas, Estadistica y Computacion, Universidad de Cantabria
Summary: The first half of this dissertation concerns the following problem: Given a lattice in R^d which refines the integer lattice Z^d, what can be said about the distribution of the lattice points inside of the half-open unit cube [0,1)^d? This question is of interest in discrete geometry, especially integral polytopes and Ehrhart theory. We observe a combinatorial description of the linear span of these points, and give a formula for the dimension of this span. The proofs of these results use methods from classical multiplicative number theory.
In the second half of the dissertation, we investigate oriented matroids from the point of view of tropical geometry. Given an oriented matroid, we describe, in detail, a polyhedral complex which plays the role of the Bergman complex for ordinary matroids. We show how this complex can be used to give a new proof of the celebrated Bohne-Dress theorem on tilings of zonotopes by zonotopes with an approach which relies on a novel interpretation of the chirotope of an oriented matroid.|
