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SUMMARY:Thomas Karam (University of  Oxford)
DTSTART:20250522T160000Z
DTEND:20250522T162500Z
DTSTAMP:20260423T041655Z
UID:CANT2025/43
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/43/
 ">After the cap-set problem\, and some properties of the slice rank</a>\nb
 y Thomas Karam (University of  Oxford) as part of Combinatorial and additi
 ve number theory (CANT 2025)\n\nLecture held in CUNY Graduate Center - Sci
 ence Center (4th floor).\n\nAbstract\nThe infamous cap-set problem asks fo
 r the size of the largest subset $A \\subset \\mathbb{F}_3^n$ not containi
 ng any solutions to the equation $x+y+z=0$ aside from the trivial solution
 s $x=y=z$. A proof that that size is bounded above by $C^n$ for some $C<3$
 \, which arose in 2016 in two breakthrough papers by Croot-Lev-Pach and by
  Ellenberg and Gijswijt (both published in the Annals of Mathematics)\, wa
 s later reformulated by Tao in a more symmetric way\, leading to the defin
 ition of a new notion of rank on tensors called the slice rank.\n\nSince t
 hen\, the slice rank has been studied further\, and the resulting properti
 es have often found related number-theoretic applications. To take the ear
 liest and perhaps simplest example\, a key component of the argument in th
 e proof of the original cap-set problem itself is that the slice rank of a
  “diagonal” tensor is equal to its number of non-zero entries\, mirror
 ing the analogous property of matrix rank.\n\nAfter reviewing some more su
 ch applications by other mathematicians\, we will present some results con
 cerning other basic properties of the slice rank\, and in particular the i
 deas behind some of their simpler proofs in the special case where the sup
 port of the tensor is contained in an antichain: there\, as established by
  Sawin and Tao\, the slice rank of the tensor is equal to the smallest num
 ber of slices that suffice to cover its support. If time allows then we wi
 ll also discuss how the proofs in this special case illuminate to some ext
 ent the proofs in the general case.\n
LOCATION:https://researchseminars.org/talk/CANT2025/43/
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