In 1969 V.Strassen realized it is possible to multiply 2x2 matrices using 7 multiplications instead of the usual 8. Since then, considerable work has been done to determine just how many scalar multiplications are necessary to execute the multiplication of nxn matrices. The problem leads to questions regarding the geometry of homogeneous spaces that are of interest in their own right. I will explain these related questions and how I solved a longstanding problem in complexity using algebraic and differential geometry.