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{\Large Department of Mathematics, BGU}

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{\Huge Colloquium}\\[0.2\baselineskip]

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\textbf{On} \emph{Tuesday, May 10, 2022}
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\textbf{At} \emph{14:30 -- 15:30}
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\textbf{In} \emph{Math -101}

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{\large\scshape Nathan Keller 
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  (BIU)
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will talk about
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{\Large\bfseries Can you hear the shape of a low-degree Boolean function?\par}
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\textsc{Abstract:}
Analysis of Boolean functions aims at ``hearing the shape'' of functions on the discrete cube \{-1,1\}\^{}n -- namely, 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'' frequencies. 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 functions on the discrete cube, 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 mean 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








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