Bounds for syzygies of monomial curves

Given a set of integers $g_0 < \ldots < g_e$, how many relations are there among them? This deceptively simple numerical question can be translated in an algebraic one, as it asks for the number of minimal generators of the defining ideal of a semigroup ring. The study of numerical properties of monoids via their associated semigroup ring (and viceversa) is a classic research topic; in this talk we explore a novel approach which extends this problem beyond the realm of semigroup rings to the one of monomial ideals, allowing to use a larger set of techniques and giving general bounds and insight.