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DTSTART:20181028T020000
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UID:calendar.64651.field_date.0@mathematics.huji.ac.il
DTSTAMP:20201201T184720Z
CREATED:20181003T040020Z
DESCRIPTION:Date: \n\n11:00am to 1:00pm\n\n\n\n\nSee also: Combinatorics\
, Seminars\, Events & SeminarsLocation: \n\nRothberg CS blgd\, room B500\,
Safra campus\, Givat\, Ram\n\n\nSpeaker: Noam Lifshitz\, BIU\nTitle: Sha
rp thresholds for sparse functions with applications to extremal combinato
rics.\nAbstract:\nThe sharp threshold phenomenon is a central topic of res
earch in the analysis of Boolean functions. Here\, one aims to give suffic
ient conditions for a monotone Boolean function $f$ to satisfy $\mu_{p}(f)
=o(\mu_{q}(f))$\, where $q = p + o(p)$\, and $\mu_{p}(f)$ is the probabili
ty that $f=1$ on an input with independent coordinates\, each taking the v
alue $1$ with probability $p$. \nThe dense regime\, where $\mu_{p}(f)=\The
ta(1)$\, is somewhat understood due to seminal works by Bourgain\, Friedgu
t\, Hatami\, and Kalai. On the other hand\, the sparse regime where $\mu_
{p}(f)=o(1)$ was out of reach of the available methods. However\, the pote
ntial power of the sparse regime was suggested by Kahn and Kalai already i
n 2006.\nIn this talk we show that if a monotone Boolean function $f$ with
$\mu_{p}(f)=o(1)$ satisfies some mild pseudo-randomness conditions then i
t exhibits a sharp threshold in the interval $[p\,q]$\, with $q = p+o(p)$.
More specifically\, our mild pseudo-randomness hypothesis is that the $p$
-biased measure of $f$ does not bump up to $\Theta(1)$ whenever we restric
t $f$ to a sub-cube of constant co-dimension\, and our conclusion is that
we can find $q=p+o(p)$\, such that $\mu_p(f)=o(\mu_q(f))$ \nAt its core\
, this theorem stems from a novel hypercontactive theorem for Boolean func
tions satisfying pseudorandom conditions\, which we call `small generalize
d influences'. This result takes on the role of the usual hypercontractivi
ty theorem\, but is significantly more effective in the regime where $p =
o(1)$. \nWe demonstrate the power of our sharp threshold result by reprovi
ng the recent breakthrough result of Frankl on the celebrated Erdos matchi
ng conjecture\, and by proving conjectures of Huang--Loh--Sudakov and Fure
di--Jiang for a new wide range of the parameters.\nBased on a joint work w
ith Keevash\, Long\, and Minzer.\n\n Export\n \n\n \nsubscribe iCal
DTSTART;TZID=Asia/Jerusalem:20181029T110000
DTEND;TZID=Asia/Jerusalem:20181029T130000
LAST-MODIFIED:20191104T100204Z
SUMMARY:Combinatorics: Noam Lifshitz\, BIU\, 'Sharp thresholds for sparse f
unctions with applications to extremal combinatorics.'
URL;TYPE=URI:https://mathematics.huji.ac.il/event/combinatorics-noam-lifshi
tz-biu-tba
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