By Sean Sather-Wagstaff (Clemson University).

Each module M over a commutative local Noetherian ring R comes equipped with certain numerical invariants, defined homologically, that allow us to detect certain structural information about the module. For example, the projective dimension detects how close M is to being projective. Not only that, but projective dimension of modules can detect whether a ring is regular, as Auslander, Buchsbaum, and Serre proved in the 1950s, and this led to the solution of a famous open question about the localization properties for regular rings. In this course, I will discuss some of these ideas (there are many out there) focusing on a few favorites, with a view toward some recent advances.