New sharp thresholds results and applications in extremal combinatorics

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 harder to apply.We will discuss new forms of this inequality and some of their consequences, such as qualitatively tight version of Bourgain's sharp threshold theorem, as well as a variant that applies for sparse families. Time permitting, we will also discuss applications to two problems in extremal combinatorics: the (cross version) of the Erdos matching conjecture, and determining the extremal size of families of vectors in {0,1,...,m-1}^n avoiding a fixed intersection size, for m\geq 3. Based on joint works with Peter Keevash, Noam Lifshitz and Eoin Long.