We develop a topological framework for proving lower bounds on sign-rank via $\mathbb{Z}_2$–equivariant topology, and use it to resolve the sign-rank of the Gap Hamming Distance problem up to lower-order terms.

For every (partial) sign matrix $A$, we associate a free $\mathbb{Z}_2$–simplicial complex $S(A)$ and show that sign-rank of $A$ ... more >>>

Given a propositional proof system $P$, we may define a formula $\text{Prf}^P_s(F)$ which is satisfiable if and only if the formula $F$ has a length $\leq s$ refutation in $P$. These formulas have received much attention in recent years due to their fundamental nature --- if a powerful proof ... more >>>

For an arbitrary family of predicates $\mathcal{F} \subseteq \{0,1\}^{[q]^k}$ and any $\epsilon > 0$, we prove a single-pass, linear-space streaming lower bound against the gap promise problem of distinguishing instances of Max-CSP$({\mathcal{F}})$ with at most $\beta+\epsilon$ fraction of satisfiable constraints from instances of with at least $\gamma-\epsilon$ fraction of satisfiable ... more >>>