This dissertation presents recent contributions to two major topics in discrete geometry: Ehrhart theory and hyperplane arrangements. In Chapter 1, which is part of joint work with Donghyun Kim and Mariel Supina, we investigate equivariant Ehrhart theory, which unites Ehrhart theory with representation theory to study the properties of convex polytopes and their symmetries. Our first contribution is to prove basic results about the equivariant Ehrhart series, which generalizes the usual Ehrhart series; in particular, we provide two rational generating functions. Secondly, we prove two original theorems for calculating the equivariant Ehrhart series using invariant triangulations. Finally, we present Sagemath code to calculate the equivariant Ehrhart series. In Chapter 2, which is joint work with Matthias Beck and Sophie Rehberg, we study rational Ehrhart theory. Our goal is to add a generating function perspective to the work of Linke, who showed that the rational and real Ehrhart counting functions of rational polytopes are quasipolynomial. We associate two rational generating functions to a rational polytope that completely describe its rational and real Ehrhart counting functions. We provide structural theorems about these generating functions: rationality, nonnegativity theorems, connections to the h*-polynomial from classical Ehrhart theory, and combinatorial reciprocity theorems. We also extend the notion of Gorenstein polytopes to the rational setting. In Chapter 3, which is joint work with Marie-Charlotte Brandenburg and Leon Zhang, we perform a computational investigation of the multivariate Ehrhart theory of polytropes using methods from toric geometry. We compute multivariate volume, Ehrhart, and h*-polynomials for all polytropes up to dimension 4, in which there are 27,248 types of maximal polytropes. We also provide a combinatorial description of the coefficients of the volume polynomials in dimension 3 in terms of regular central subdivisions of the fundamental polytope. In Chapter 4, which is joint work with Jean-Philippe Labbé and Michael Cuntz, we study simplicial hyperplane arrangements. The number of isomorphism classes of rank 3 simplicial hyperplane arrangements is unknown. We provide a Sagemath package containing the most up to date catalogue of simplicial hyperplane arrangements of rank 3, giving normals and invariants. We add structure to this catalogue by classifying the arrangements according to whether their associated lattices of regions are always, sometimes, or never congruence normal depending on the choice of base region. Our novel methods for checking congruence normality work in any dimension and make use of the oriented matroid associated to a hyperplane arrangement. We prove that finite Weyl groupoids of any dimension have congruence normal lattices of regions.
