Gil Kalai: Analysis of Boolean functions some results and two problems

A few results and two general conjectures regarding analysis of Boolean functions, influence, and threshold phenomena will be presented. 

Boolean functions are functions of n Boolean variables with values in {0,1}. They are important in combinatorics, theoretical computer science, probability theory, and game theory. 

Influence.  Causality is a topic of great interest in statistics, physics, philosophy, law, economics, and many other places.  If causality is not complicated enough, we can ask what is the influence one event has on another one. Ben-Or and Linial  1985 paper studied influence in the context of collective coin flipping - a problem in theoretical computer science.

Fourier. Over the last two decades, Fourier analysis of Boolean functions and related objects played a growing role in discrete mathematics, and theoretical computer science.

Threshold phenomena. Threshold phenomena refer to sharp transition in the probability of certain events  depending on a parameter p near a critical value. A classic example that goes back to Erdős and Rényi, is the behavior of certain monotone properties of random graphs.

Influence of variables on Boolean functions is connected to their Fourier analysis and threshold behavior, as well as to discrete isoperimetry and noise sensitivity. 

The first Conjecture to be described (with Friedgut) is called the Entropy-Influence Conjecture. (It was featured on Terry Tao's blog.) It gives a far reaching extention to the KKL theorem, and theorems by Friedgut, Bourgain, and me.

The second Conjecture (with Kahn) proposes a far-reaching generalization to results by Friedgut, Bourgain and Hatami. 

We also mention a simple yet annoying question related to Kruskal Katona's theorem.