Year of Publication

2015

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department

Mathematics

First Advisor

Dr. Benjamin Braun

Second Advisor

Dr. Carl Lee

Abstract

An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal h*-vector.Although various sufficient conditions have been found, necessary conditions remain a challenge. Highly-structured polytopes, such as the polytope of real doubly-stochastic matrices, have been proven to possess unimodal h*-vectors, but the same is unknown even for small variations of it.

In this dissertation, we mainly consider two particular classes of polytopes: reflexive simplices and the polytope of symmetric real doubly-stochastic matrices. For the first class, we discuss an operation that preserves reflexivity, integral closure, and unimodality of the h*-vector, providing one explanation for why unimodality occurs in this setting. We also discuss the failure of proving unimodality in this setting using weak Lefschetz elements. With the second class, we prove partial unimodality results by examining their toric ideals and using a correspondence between these and regular triangulations of the polytopes. Lastly, we describe the computational methods used to help develop these results. Several software programs were used, and the code has proven useful outside of the main focus of this work.

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