Title: Advances in algebraic multigrid for topological-based problems.

Differential k-forms arise in scientific disciplines ranging from
electromagnetics to computer graphics. Applications include stable
eddy-current simulations, topological analysis of sensor network coverage,
visualization techniques for fluid flow, and construction of tangent vector
fields in graphics processing. In this talk we consider so called Laplace-de
Rham problems that arise in many of these examples and focus on the solution of
the discrete k-forms. In particular, we highlight new aggregation-based
algebraic multigrid approaches to the problem that generalize to high
dimension.

The talk will outline the basics of the k-form Laplacian and offer a
description of our algebraic approach to the solution. We draw a connection
between the coarse problems constructed in the multigrid algorithm and the
familiar de Rham complex at the fine level. The numerical evidence of the
efficiency and generality of the method is presented.  Extensions of the
approach to more compicated problems are also highlighted, along with the
necessary ingredients for a scalable multigrid algorithm.