We study the fundamental challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over $\mathbb{F}_{2}$. Our main contributions include:

1. In STOC 2020, CHHLZ introduced a new technique to prove correlation bounds. Using their technique they established new correlation bounds for low-degree polynomials. They conjectured that their ... more >>>

Let $G$ be a group such that any non-trivial representation has dimension
at least $d$. Let $X=(X_{1},X_{2},\ldots,X_{t})$ and $Y=(Y_{1},Y_{2},\ldots,Y_{t})$
be distributions over $G^{t}$. Suppose that $X$ is independent from
$Y$. We show that for any $g\in G$ we have
\[
\left|\mathbb{P}[X_{1}Y_{1}X_{2}Y_{2}\cdots X_{t}Y_{t}=g]-1/|G|\right|\le\frac{|G|^{2t-1}}{d^{t-1}}\sqrt{\mathbb{E}_{h\in G^{t}}X(h)^{2}}\sqrt{\mathbb{E}_{h\in G^{t}}Y(h)^{2}}.
\]
Our results generalize, improve, and ... more >>>

We make progress on understanding a lower bound technique that was recently used by the authors to prove the first superpolynomial constant-depth circuit lower bounds against algebraic circuits.

More specifically, our previous work applied the well-known partial derivative method in a new setting, that of 'lopsided' set-multilinear polynomials. A ... more >>>