This note is an exposition of the recent preprints from OpenAI that establish the bound $\omega \leq 9/4$ on the matrix multiplication exponent. We argue that the heart of these preprints is a new technique, which we formulate as the untwisting lemma for semidirect sums of tensors. This gives a ... more >>>

We study depth-$4$ algebraic circuits with bounded top and bottom fan-in over fields of characteristic $0$. These circuits compute sums of at most $k$ products of polynomials of degree at most $\delta$. We denote this class of circuits by $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$. Our hard polynomial for this class is $P_{k + 1,n}$, ... more >>>

A boolean function $f$ has Fourier dimension $k$ if its strings with nonzero Fourier coefficients span a subspace of dimension $k$. We first consider the property testing task of determining whether a function has Fourier dimension at most $k$, or is $\epsilon$-far from being so. We show that there is ... more >>>