  We analyze tensors in the tensor product of three m-dimensional vector spaces
satisfying Strassen's equations for border rank m. Results include: two purely
geometric characterizations of the Coppersmith-Winograd tensor, a reduction to
the study of symmetric tensors under a mild genericity hypothesis, and numerous
additional equations and examples. This study is closely connected to the study
of the variety of m-dimensional abelian subspaces of the space of endomorphisms
of an m-dimensional vector space, and the subvariety consisting of the Zariski
closure of the variety of maximal tori, called the variety of reductions.
