Numerical semigroups : insights into minimal resolutions and wilf's conjecture

Numerical Semigroups appear across many areas of mathematics, including algebraic geometry, convex geometry, number theory, integer programming, and even music theory. They are versatile and adaptable as one can use the combinatorics of these semigroups to study otherwise hard-to-compute algebraic properties. This thesis explores numerical semigroups from three perspectives. Firstly, we address a conjecture of Wilf arising from the money-changing problem: we prove that the conjectured inequality is preserved under gluing-a way to combine semigroups. Secondly, we study Kunz-Waldi semigroups, building up from two coprime numbers p < q: we characterize (a) their principal matrices (square matrices that contain intrinsic information about the semigroups themselves) and (b) those KW semigroups whose defining ideal is determinantal. We next identify KW semigroups that lie on the interior of the same face of the Kunz cone as those in (b), thus, providing explicit Betti numbers and minimal resolutions of all those KW semigroups. Thirdly, we study Sally type semigroups: S = ⟨[e, 2e−1]\{e+j1, e+j2, ..., e+jk}, 1 < j1 < ... < jk < e−1⟩. When k = 1, 2, we give formulas for the Frobenius number and type, and describe the defining ideal of k[S] using Hochster's formula to first compute the number of minimal generators. When k > 1 and ji's consecutive, we characterize when they are symmetric and construct minimal resolutions for these Gorenstein semigroups rings.