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