  Until a few years ago, the fastest known matrix multiplication algorithm, due
to Coppersmith and Winograd (1990), ran in time $O(n^{2.3755})$. Recently, a
surge of activity by Stothers, Vassilevska-Williams, and Le Gall has led to an
improved algorithm running in time $O(n^{2.3729})$. These algorithms are
obtained by analyzing higher and higher tensor powers of a certain identity of
Coppersmith and Winograd. We show that this exact approach cannot result in an
algorithm with running time $O(n^{2.3725})$, and identify a wide class of
variants of this approach which cannot result in an algorithm with running time
$O(n^{2.3078})$; in particular, this approach cannot prove the conjecture that
for every $\epsilon > 0$, two $n\times n$ matrices can be multiplied in time
$O(n^{2+\epsilon})$.
  We describe a new framework extending the original laser method, which is the
method underlying the previously mentioned algorithms. Our framework
accommodates the algorithms by Coppersmith and Winograd, Stothers,
Vassilevska-Williams and Le Gall. We obtain our main result by analyzing this
framework. The framework is also the first to explain why taking tensor powers
of the Coppersmith-Winograd identity results in faster algorithms.
