Докладчиком предлагается рассмотреть две темы. Распределение времени семинара между темами будет определяться пожеланиями слушателей семинара.

Title: A proof of the Sensitivity Conjecture

In this talk we will show that for any function f block sensitivity of f is bounded by a polynomial of sensitivity of f.

Based on paper of Hao Huang: "Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture".

Title: A tradeoff between length and width in PCR

For various proof systems we know that proof of small size can be translated into a proof of small degree (or small width). In this constructions, the size of the final proof is usually exponential in terms of variables of the formula. For Resolution Thapen showed that it is necessary. We extend his result into polynomial calculus proof system.

More formally:
We construct a formula \varphi of size n that in polynomial calculus resolution (PCR) has a refutation of polynomial size and degree c := n^{1 - o(1)} and hence a refutation of exponential size and degree n^{0.5 + o(1)}. However, we show that there is no refutation achieving both polynomial size and degree at most c - O(\lg n)$ at the same time.

Joint work with Guillaume Lagarde, Jakob Nordström, Joseph Swernofsky.