DMS Colloquium: Mark Walker

Speaker: Mark Walker (Willa Cather Professor, University of Nebraska--Lincoln)

Title: The Total and Toral Rank Conjectures

Abstract: Assume $X$ is a nice topological 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 such a space $X$ by gluing them together.

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 $M$.

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

Faculty host: Michael Brown