In 1969 Strassen showed the standard algorithm for multiplying nxn matrices is not optimal. Since then there has been intense interest in determining just how fast matrices can be multiplied. It is even conjectured that asymptotically, as n goes to infinity, one can multply nxn matrices nearly as fast as one can add them. I will describe a new approach to this problem via symmetry (representation theory).
This is joint work with G. Ballard, A. Conner, J. Hauenstein, C. Ikenmeyer and N. Ryder.