Each numerical semigroup with smallest positive element \(m\) can be represented as an integer point in a polyhedral cone \(C_m\), known as the Kunz cone. The faces of this cone organize numerical semigroups into different layers, and this organization reflects several important properties of the semigroups. In this series of talks, we discuss how this structure influences the behavior of minimal free resolutions of Apéry toric ideals. Furthermore, we explain how the infinite free resolution of the ground field over a semigroup algebra follows this same pattern, providing a combinatorial approach to understanding properties such as Golodness and the rationality of the Poincaré series.In the first talk, we will focus on the basic theory of the Kunz polyhedron and numerical semigroups required for the subsequent talks.
