This talk is part of the Advances in Boolean Function Analysis Lecture Series. The series will feature weekly two-hour lectures that aim to address both the broad context of the result and the technical details. Though closely related in theme, each lecture will be self-contained. Join us weekly at 10:00 a.m. PDT, from July 15, 2020 to August 18, 2020. There is a five minute break at the end of the first hour.

Abstract:
We revisit several classical inequalities which relate the influences of a Boolean function to its variance - the Kahn-Kalai-Linial (KKL) inequality and its generalizations by Friedgut and Talagrand, and the relation between influences and noise stability by Benjamini-Kalai-Schramm. We will introduce a new method towards the proofs of these inequalities (based on stochastic calculus and the analysis of jump processes). Our method resolves a '96 conjecture of Talagrand, deriving a bound which strengthens both Talagrand's sensitivity inequality and the KKL inequality. Our method also produces robust versions of some of the aforementioned bounds. Joint work with Renan Gross.