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SUMMARY:Alexey Balitskiy (IAS Princeton\, and Institute for Information Tr
 ansmission Problems RAS)
DTSTART:20220518T121000Z
DTEND:20220518T130000Z
DTSTAMP:20260423T022714Z
UID:GDS/59
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GDS/59/">Sys
 tolic freedom and rigidity modulo 2</a>\nby Alexey Balitskiy (IAS Princeto
 n\, and Institute for Information Transmission Problems RAS) as part of Ge
 ometry and Dynamics seminar\n\n\nAbstract\nThe $k$-dimensional systole of 
 a closed Riemannian $n$-dimensional manifold $M$ \nis the infimal $k$-volu
 me of a non-trivial $k$-cycle (with some coefficients). \nIn '90s\, Gromov
  asked if the product of the $k$-systole and the $(n-k)$-systole \nis boun
 ded from above by the volume of $M$ (up to a dimensional factor)\; this \n
 would manifest the \\emph{systolic rigidity}. Freedman exhibited the first
  \nexamples with $k=1$ and mod 2 coefficients where this fails\; this mani
 fests \nthe \\emph{systolic freedom}. In a joint work in progress with Han
 nah Alpert \nand Larry Guth\, we show that Freedman's examples are almost 
 as "free" as \npossible\, and the systolic rigidity almost holds\, with $k
 =1$ and mod 2 \ncoefficients. Namely\, on a manifold of bounded local geom
 etry\, \n$\\mbox{systole}_1(M) \\cdot \\mbox{systole}_{n-1}(M) \\le c_\\ep
 silon \\mbox{volume}(M)^{1+\\epsilon}$\, \nas long as the left-hand side i
 s finite ($H_1(M\; \\mathbb{Z}/2)$ is non-trivial). \nThe proof\, which I 
 will explain\, is based on the Schoen--Yau--Guth--Papasoglu \nminimal surf
 ace method.\n
LOCATION:https://researchseminars.org/talk/GDS/59/
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