  The complexity of matrix multiplication is measured in terms of $\omega$, the
smallest real number such that two $n\times n$ matrices can be multiplied using
$O(n^{\omega+\epsilon})$ field operations for all $\epsilon>0$; the best bound
until now is $\omega<2.37287$ [Le Gall'14]. All bounds on $\omega$ since 1986
have been obtained using the so-called laser method, a way to lower-bound the
`value' of a tensor in designing matrix multiplication algorithms. The main
result of this paper is a refinement of the laser method that improves the
resulting value bound for most sufficiently large tensors. Thus, even before
computing any specific values, it is clear that we achieve an improved bound on
$\omega$, and we indeed obtain the best bound on $\omega$ to date: $$\omega <
2.37286.$$ The improvement is of the same magnitude as the improvement that [Le
Gall'14] obtained over the previous bound [Vassilevska W.'12]. Our improvement
to the laser method is quite general, and we believe it will have further
applications in arithmetic complexity.
