Caltech/UCLA Joint Analysis Seminar

PLEASE NOTE DIFFERENT TIME

Analysis of Boolean functions studies functions on the discrete cube {-1,1}n, aiming at understanding what the structure of the (discrete) Fourier transform tells us about the function. In this talk we focus on the structure of low-degree functions on the discrete cube, namely, on functions whose Fourier coefficients are concentrated on low degrees. While such functions look very simple, we are surprisingly far from understanding them well, even in the most basic first-degree case.

We shall present several results on first-degree Boolean functions, including the recent proof of Tomaszewski's conjecture (1986) which asserts that any first-degree function (viewed as a random variable) lies within one standard deviation from its expectation with probability at least 1/2. Then we shall discuss several core open questions, which boil down to understanding, what does the knowledge that a low-degree function is bounded, or is two-valued, tell us about its structure.

Based on joint work with Ohad Klein