This dissertation explores problems of convexity, combinatorics, and algebra associated with semigroups of polyhedral lattice points.

In \Cref{ChapterColored}, we first attempt to generalize and extend three well-known convexity theorems, including Helly theorem, Tverberg theorem, and Colorful Carath eodory theorem, to affine semigroups. We define a novel notion, chromatic representations of semigroup elements, this is in contrast to the colorful theory developed by B ar any et al. Later, we focus on one-dimensional affine semigroups, numerical semigroups, and study the number of chromatic solutions in numerical semigroups.

In \Cref{ChapterWeighted}, we generalize the classical Hilbert functions and Hilbert series of a semigroup algebra to have weightings. We list three ways to add weightings, $q$-weighting, $r$-weighting, and $s$-weighting, and study their relationships. We find that the $q$-weighting can derive other weightings. Later, we specialize to the special family of semigroup algebras, the Ehrhart rings. We study and extend the properties of $h^{*}$-nonnegativity and Ehrhart–Macdonald reciprocity for the Ehrhart series under these three weightings.

In \Cref{ChapterEhrhart}, we focus on the Ehrhart functions under the $s$-weighting and give a practical method to evaluate the $s$-weighted Ehrhart function. Specifically, we construct a new polytope, the weight-lifting polytope, and build a connection between the $s$-weighted Ehrhart function and the classical Ehrhart function. Later, we present several applications and experiments of our method in combinatorial representation theory and number theory.

In \Cref{ChapterKakeya}, we discuss a long-standing conjecture, Kakeya's conjecture, which brings a surprising connection between numerical semigroups and symmetric polynomials. We give partial results, prove the conjecture for two variables, and outline a general computer proof for an arbitrary number of variables.