Free resolutions, syzygies, regularity and Betti numbers

In this preparatory lecture I will introduce and state the fundamental properties of some objects and invariants relevant in the study of a module using free resolutions. We will start from the first appearance of these ideas in the work of Hilbert (and even earlier of Cayley), until the recently introduced Boij-Soderberg theory. Time permitting, we will also discuss briefly some bounds and conjectures on the aformentioned invariants.

This is a preparatory lecture for the ICM talk of Irena Peeva.