(1) We present algorithms for all dense linear algebra (solving systems of linear equations, least squares, eigenvalue problems) that use asymptotically as few arithmetic operations as the fastest matrix multiplication routine that will ever be invented. Furthermore, these algorithms will be numerically stable, even if the fastest matrix multiplication routine is not. This is joint work with Ioana Dumitriu, Olga Holtz and Robert Kleinberg.

(2) We present algorithms for iteratively solving sparse linear systems with algorithms that minimize the latency and bandwidth costs. On a parallel machine, this means network latency between processors, and on a sequential machine this means latency and bandwidth between levels of a memory hierarchy. This is joint work with Mark Hoemmen, Marghoob Mohiyuddin and Kathy Yelick

(3) We present algorithms for dense and sparse QR and LU decompositions that similarly minimize latency and bandwidth costs. This is joint work with Laura Grigori, Mark Hoemmen, Julien Langou and Jessica Schoen.