Dor Minzer, Institute for Advanced Study

Abstract

The classical hypercontractive inequality for the Boolean hypercube lies at the core of many results in analysis of Boolean functions. Though extensions of the inequality to different domains (e.g. the biased hypercube) are known, they are often times quantitatively weak, making them hard to apply.

We will discuss new forms of this inequality and some of their consequences, such as quantitatively tight version of Bourgains sharp threshold theorem and sharp threshold results for sparse families. Time permitting, we will also discuss applications to two problems in extremal combinatorics: the Erdos matching conjecture, and families avoiding a fixed intersection in the multi-cube, {0,1,...,m-1}^n, for m>=3.

Based on joint works with Peter Keevash, Noam Lifshitz and Eoin Long.