Jonah Blasiak

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 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.

The toric ideal of a graphic matroid is generated by quadrics

Describing minimal generating sets of toric ideals is a well-studied and difficult problem. Neil White conjectured in 1980 that the toric ideal associated to a matroid is generated by quadrics corresponding to single element symmetric exchanges. We give a combinatorial proof of White’s conjecture for graphic matroids.

Geometric complexity theory IV : nonstandard quantum group for the Kronecker problem

Introduction Basic concepts and notation Hecke algebras and canonical bases The quantum group

Noncommutative Schur functions, switchboards, and Schur positivity

GL_q(V)

Kronecker coefficients and noncommutative super Schur functions

Bases for

Kronecker coefficients for one hook shape

GL_q(V)

A special case of Hadwiger's conjecture

modules Quantum Schur-Weyl duality and canonical bases Notation for

Haglund’s conjecture on 3-column Macdonald polynomials

GL_q(V) \times GL_q(W)

Cyclage, catabolism, and the affine Hecke algebra

The nonstandard coordinate algebra

A factorization theorem for affine Kazhdan-Lusztig basis elements

\mathscr{O}(M_q(\check{X}))