We prove that the most natural low-degree test for polynomials over finite fields is ``robust'' in the high-error regime for linear-sized fields. Specifically we consider the ``local'' agreement of a function $f\colon \mathbb{F}_q^m \to \mathbb{F}_q$ from the space of degree-$d$ polynomials, i.e., the expected agreement of the function from univariate ... more >>>

Can every $n$-bit boolean function with deterministic query complexity $k\ll n$ be restricted to $O(k)$ variables such that the query complexity remains $\Omega(k)$? That is, can query complexity be condensed via restriction? We study such hardness condensation questions in both query and communication complexity, proving two main results.

$\bullet~$ $\mathbf{Negative}$: ... more >>>

We revisit the fundamental Boolean Matrix Multiplication (BMM) problem. With the invention of algebraic fast matrix multiplication over 50 years ago, it also became known that BMM can be solved in truly subcubic $O(n^\omega)$ time, where $\omega<3$; much work has gone into bringing $\omega$ closer to $2$. Since then, a ... more >>>