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UID:www.tcs.tifr.res.in/event/1076
DTSTAMP:20230914T125949Z
SUMMARY:Identifying some necessary conditions for a function to be Boolean
DESCRIPTION:Speaker: Nikhil Mande (Georgetown University\, Washington\, D.C
 .)\n\nAbstract: \nThe seminar will happen via Google meet.\n\nAbstract:  E
 very Boolean function $f : \\mathbb{F}_2^n \\to \\{-1\, 1\\}$ has a unique
  Fourier representation\, that is\, $f = \\sum_{\\alpha \\in \\mathbb{F}_2
 ^n}\\hat{f}(\\alpha)\\chi_{\\alpha}$. One can obtain the following identit
 ies using elementary techniques.\n\n1) Parseval's identity: $\\sum_{\\alph
 a \\in \\mathbb{F}_2^n}\\hat{f}(\\alpha)^2 = 1$.\n2) For all $\\gamma \\ne
 q 0^n$\, we have $\\sum_{(\\alpha_1\, \\alpha_2) \\in \\mathbb{F}_2^n \\ti
 mes \\mathbb{F}_2^n : \\alpha_1+\\alpha_2=\\gamma}\\hat{f}(\\alpha_1)\\hat
 {f}(\\alpha_2)=0$. Henceforth\, we refer to these conditions as Titsworth'
 s conditions.\n\nLet $\\mathcal{S}$ denote the Fourier support of $f$\, th
 at is\, the set of elements of $\\mathbb{F}_2^n$ that have non-zero Fourie
 r coefficients. Titsworth's conditions can be seen to imply that for every
  pair $(\\alpha_1\, \\alpha_2)$ of elements in $\\mathcal{S}$\, there must
  exist at least one other pair $(\\beta_1\, \\beta_2)$ of elements in $\\m
 athcal{S}$ such that $\\alpha_1 \\oplus \\alpha_2 = \\beta_1 \\oplus \\bet
 a_2$. We investigate whether this condition in itself is sufficient for $\
 \mathcal{S}$ to be the Fourier support of a Boolean function.\n\nOnly basi
 c mathematical familiarity will be assumed. I will cover the required prer
 equisites from Fourier analysis of Boolean functions in the beginning of t
 he talk.\n\nBased on joint work with Swagato Sanyal (IIT Kharagpur). Link 
 to full paper: https://arxiv.org/abs/2008.00266\n
URL:https://www.tcs.tifr.res.in/web/events/1076
DTSTART;TZID=Asia/Kolkata:20200815T110000
DTEND;TZID=Asia/Kolkata:20200815T120000
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