We show that there are degree-$d$ polynomials over $\mathbb{F}_{2}$ with
correlation $\Omega(d/\sqrt{n})$ with the majority function on $n$
bits. This matches the $O(d/\sqrt{n})$ bound by Smolensky.

more >>>

We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense ... more >>>

We revisit the fundamental problem of determining seed length lower bounds for strong extractors and natural variants thereof. These variants stem from a ``change in quantifiers'' over the seeds of the extractor: While a strong extractor requires that the average output bias (over all seeds) is small for all input ... more >>>