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SUMMARY:GS Advanced Topics in PDE: The Boolean surface area of polynomial 
 threshold function [Oberseminar Analysis]
DTSTART:20260424T121500Z
DTEND:20260424T131500Z
DTSTAMP:20260421T152900Z
UID:indico-event-1250@math-events.uni-bonn.de
DESCRIPTION:Speakers: Alexander Volberg (MSU und Bonn)\n\nPolynomial thres
 hold functions (PTFs) are an important low-complexity class of Boolean fun
 ctions\, with strong connections to learning theory and approximation theo
 ry. Recent work on learning and testing PTFs has exploited structural and 
 isoperimetric properties of the class\, especially bounds on average sensi
 tivity\, one of the central themes in the study of PTFs since the Gotsman
 –Linial conjecture. In this work we exhibit a new geometric sense in whi
 ch PTFs are tightly constrained\, by studying them through the lens of the
  \\textit{Boolean surface area} (or Talagrand boundary):\n \\[ BSA[f] = E
 |\\nabla f| = E \\sqrt{Sens_f(x)}\, \\]\nwhich is a natural measure of ver
 tex-boundary complexity on the discrete cube. Our main result is that ever
 y degree-$d$ PTF $f$ has subpolynomial Boolean surface area:\n \\[ BSA[f]
  \\le exp(C(d)\\sqrt{logn}). \\]\nThis is a superpolynomial improvement ov
 er the previous bound of $n^{1/4}(log n)^{C(d)}$ that follows from Kane's 
 landmark bounds on average sensitivity of PTFs [?].\nDegree-$d$ PTFs thus 
 satisfy a stronger form of geometric regularity than was previously visibl
 e from influence bounds alone. As an application\, we obtain improved nois
 e sensitivity estimates in the case of small noise parameter. \nPlease al
 so find the pdf file for the abstract here. \n\nhttps://math-events.uni-b
 onn.de/event/1250/
LOCATION:Endenicher Allee 60\, Seminarraum 0.011 (Mathezentrum)
URL:https://math-events.uni-bonn.de/event/1250/
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