How quickly can one multiply two large matrices? Surprisingly, this natural question has never been completely answered. In 1969 Volker Strassen showed how to multiply n-by-n matrices in roughly n^2.81 operations, which was an amazing improvement over the obvious n^3 algorithm; currently the best method known (discovered by Coppersmith and Winograd in 1986) reaches exponent 2.376. Chris Umans and I proposed using the representation theory of finite groups to develop fast algorithms. Together with Bobby Kleinberg and Balazs Szegedy, we made some progress in this direction, and we have formulated several conjectures that would achieve exponent 2.
This talk will describe the framework, discuss what we can prove, and explain what we wish we could prove.