Tensors with continuous symmetry

Computer scientists have conjectured that it is nearly as easy to multiply large matrices as it is to add them. They define the exponent of matrix multiplication $\omega$ to be the infimum of the numbers $\tau$ such that $n\times n$ matrices may be multiplied using $O(n^{\tau })$ arithmetic operations. The conjecture is that $\omega =2$. The problem was posed in 1969 and there was steady progress on proving upper bounds for $\omega$ that ended in 1989. As a first attempt to unblock research on the exponent, I will discuss variants of the conjecture from a geometric perspective. Independent of matrix multiplication, it leads to new, previously uninvestigated properties of tensors of interest in their own right.

This is joint work with A. Conner, F. Gesmundo, E. Ventura and Y. Wang.