## David Eisenbud

Geoffrey Horrocks, for reasons related to the problem of constructing vector bundles of small rank on projective space; and David Buchsbaum and I, on the basis of many computations, made a conjecture in the1970’s about the sum of the ranks of the free modules in a free resolution. Now Mark Walker (U. Nebraska, Lincoln) has give a beautiful, simple proof, bringing in just one new idea, from K-theory. In this talk, I’ll explain the conjecture and Walker’s proof and the new questions it suggests.

## Jake Levinson

Boij-Söderberg theory is a structure theory for syzygies of graded modules: a near-classification of the possible Betti tables of such modules (these tables record the degrees of generators in a minimal free resolution). One of the surprises of the theory was the discovery of a "dual" classification of sheaf cohomology tables on projective space. I'll tell part of this story, then describe some recent extensions of it to the setting of Grassmannians. Here, the algebraic side concerns modules over a polynomial ring in $kn$ variables, thought of as the entries of a $k \times n$ matrix. The goal is to classify "GLk-equivariant Betti tables", recording the syzygies of equivariant modules, and relate them to sheaf cohomology tables on the Grassmannian $Gr(k,n)$. This work is joint with Nic Ford and Steven Sam.