Georgia Institute of Technology College of Sciences

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 everywhere, and if causality is not complicated enough, we can ask what is the influence one event has on another one. Ben-Or and Linial studied influence in the context of collective coin flipping---a problem in theoretical computer science. Fourier analysis. 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 Erdos and Renyi, 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 Tao's blog.) It gives a far-reaching extension to the KKL theorem, and theorems by Friedgut, Bourgain, and the speaker. The second Conjecture (with Kahn) proposes a far-reaching generalization of results by Friedgut, Bourgain and Hatami.