We present a "group-theoretic" approach to producing fast algorithms for
matrix multiplication. In this framework, one devises algorithms by
constructing non-abelian groups with certain properties. The algorithms
themselves are natural and essentially amount to taking the Discrete Fourier
Transform over these groups.

We construct several families of groups that achieve matrix multiplication
exponent significantly less than 3 (but not beating the current best of
2.376...). This leads to two appealing conjectures, one combinatorial and
the other algebraic. Either one would imply that the exponent of matrix
multiplication is 2.

This is joint work with Henry Cohn, Bobby Kleinberg, and Balazs Szegedy.

The talk is based on two papers:

1. "A group-theoretic approach to fast matrix multiplication" (FOCS 2003)
http://front.math.ucdavis.edu/math.GR/0307321

2. "Group-theoretic algorithms for matrix multiplication" (FOCS 2005)
http://www.cs.caltech.edu/~umans/papers/CKSU05.pdf