Title: The Total and Toral Rank Conjectures

Abstract: Assume X is a nice space (a compact CW complex) that admits a fixed-point free action by a d-dimensional torus T. For example, X could be T acting on itself in the canonical way. The Toral Rank Conjecture, due to Halperin, predicts that the sum of the (topological) Betti numbers of X must be at least 2^d. Put more crudely, this conjecture predicts that it takes at least 2^d cells to build X by gluing.

Now suppose M is a module over the polynomial ring k[x_1, \dots, x_d] that is finite dimensional as a k-vector space. The Total Rank Conjecture, due to Avramov, predicts that the sum of algebraic Betti numbers of M must be at least 2^d. Here, the algebraic Betti numbers refer to the ranks of the free modules occurring in the minimal free resolution of M. The Total Rank Conjecture is a weak form of the well-known Buchsbaum-Eisenbud-Horrocks Conjecture.

In this talk I will discuss the relationship between these conjectures and recent progress toward settling them.