Chris Umans (CalTech)
Monday, June 5, 2017, 2:00pm
Room: CSE 4258
Title: On cap sets and the group-theoretic approach to matrix multiplication
In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent 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 exponent two in this framework. We connect the effort to prove exponent two via this group-theoretic approach to the existence of large "tricolored sum-free sets" in associated finite groups. We then rule out obtaining exponent two from abelian groups of bounded exponent.
I'll describe this connection, the recent breakthrough of Croot, Lev, Pach, Ellenberg, and Gijswijt resolving the Cap Set Conjecture, and our extension of this result to abelian groups of bounded exponent, which then yields our main result.
Joint work with Jonah Blasiak, Tom Church, Henry Cohn, Josh Grochow, Eric Naslund, and Will Sawin.