Time: 9:30 -- 11: 00, July 16, 2025

Venue: Room 612, A6, Institute of Mathematics-VAST

Abstract: The talk is meant to serve as background for the talks given by D. Eisenbud and B. Ulrich at the conference in memoriam of Jurgen Herzog at VIASM the week after. Let (R,m,k) be a commutative local ring. There has been a long tradition of trying to study R via how certain classes of modules decompose into indecomposable objects (this is the representation theory part). On the other hand, another significant area of research focuses on the ranks of free modules appearing in minimal resolution of say, the residue field k (the Betti numbers of k). Recently, works by the author and others have discovered interesting patterns on the indecomposable objects in the syzygies of k and sometimes of all modules over R. This has led to new knowledge about old rings, such as Golod rings, but also to new classes of rings.  The talk will introduce classical theory and give a broad overview of recent results.