Title: Geometry and the complexity of matrix multiplication

Abstract: In 1968 V. Strassen discovered that the usual row-column method for multiplying matrices is not optimal. After much work, it is now generally conjectured that as the size of the matrices grows large, it becomes nearly as easy to multiply two matrices as it is to add them! I will give a history of this astounding conjecture.

It has been approached using methods from combinatorics, probability, statistical mechanics, and other areas. I will primarily discuss how the conjecture is naturally approached as a problem in algebraic geometry and representation theory.