Advances in Boolean Function Analysis — KKL via Random Restrictions

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 give alternate proofs for three related results in analysis of Boolean functions, namely the KKL Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a new approach: we look at the first Fourier level of the function after a suitable random restriction and we apply the Log-Sobolev inequality appropriately. In particular, we avoid using the hypercontractive inequality that is common to the original proofs. Our proofs might serve as an alternative, uniform exposition to these theorems and the techniques might benefit other research. 

In this talk we will focus on our techniques and along the way we will prove the KKL Theorem using our new approach.