Let $k=k(n)$ be the largest integer such that there
exists a $k$-wise uniform distribution over $\zo^n$ that
is supported on the set $S_m := \{x \in \zo^n : \sum_i
x_i \equiv 0 \bmod m\}$, where $m$ is any integer. We
show that $\Omega(n/m^2 \log m) \le k \le 2n/m + ... more >>>

We survey recent progress in the proof complexity of strong proof systems and its connection to algebraic circuit complexity, showing how the synergy between the two gives rise to new approaches to fundamental open questions, solutions to old problems, and new directions of research. In particular, we focus on tight ... more >>>

We consider depth 2 unbounded fan-in circuits with symmetric and linear threshold gates. We present a deterministic algorithm that, given such a circuit with $n$ variables and $m$ gates, counts the number of satisfying assignments in time $2^{n-\Omega\left(\left(\frac{n}{\sqrt{m} \cdot \poly(\log n)}\right)^a\right)}$ for some constant $a>0$. Our algorithm runs in time ... more >>>