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On cap sets and the group-theoretic approach to matrix multiplication
Discrete analysis (2017) Cet article a éte moissonné depuis la source Scholastica

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In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent $ω$ of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans proposed specific conjectures for how to obtain $ω=2$. In this paper we rule out obtaining $ω=2$ in this framework from abelian groups of bounded exponent. To do this we bound the size of tricolored sum-free sets in such groups, extending the breakthrough results of Croot, Lev, Pach, Ellenberg, and Gijswijt on cap sets. As a byproduct of our proof, we show that a variant of tensor rank due to Tao gives a quantitative understanding of the notion of unstable tensor from geometric invariant theory.
Publié le : 
@article{DAS_2017_a17,
     author = {Jonah Blasiak and Thomas Church and Henry Cohn and Joshua A. Grochow and Eric Naslund and William F. Sawin and Chris Umans},
     title = {On cap sets and the group-theoretic approach to matrix multiplication},
     journal = {Discrete analysis},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2017_a17/}
}
TY  - JOUR
AU  - Jonah Blasiak
AU  - Thomas Church
AU  - Henry Cohn
AU  - Joshua A. Grochow
AU  - Eric Naslund
AU  - William F. Sawin
AU  - Chris Umans
TI  - On cap sets and the group-theoretic approach to matrix multiplication
JO  - Discrete analysis
PY  - 2017
UR  - http://geodesic.mathdoc.fr/item/DAS_2017_a17/
LA  - en
ID  - DAS_2017_a17
ER  - 
%0 Journal Article
%A Jonah Blasiak
%A Thomas Church
%A Henry Cohn
%A Joshua A. Grochow
%A Eric Naslund
%A William F. Sawin
%A Chris Umans
%T On cap sets and the group-theoretic approach to matrix multiplication
%J Discrete analysis
%D 2017
%U http://geodesic.mathdoc.fr/item/DAS_2017_a17/
%G en
%F DAS_2017_a17
Jonah Blasiak; Thomas Church; Henry Cohn; Joshua A. Grochow; Eric Naslund; William F. Sawin; Chris Umans. On cap sets and the group-theoretic approach to matrix multiplication. Discrete analysis (2017). http://geodesic.mathdoc.fr/item/DAS_2017_a17/