Workshop on Additive Combinatorics and Algebraic Connections
Quantitative Inverse Theorem for Gowers Uniformity Norms $\mathsf{U}^5$ and $\mathsf{U}^6$ in $\mathbb{F}_2^n$
Abstract: In this talk, I will discuss a proof of a quantitative version of the inverse theorem for Gowers uniformity norms $\mathsf{U}^5$ and $\mathsf{U}^6$ in $\mathbb{F}_2^n$. The proof starts from an earlier partial result of Gowers and myself which reduces the inverse problem to a study of algebraic properties of certain multilinear forms. The most of the argument is a study of the relationship between the natural actions of $\operatorname{Sym}_4$ and $\operatorname{Sym}_5$ on the space of multilinear forms and the partition rank, using an algebraic version of regularity method. One of the outcomes of the proof is a positive answer to a conjecture of Tidor about approximately symmetric multilinear forms in 5 variables, which is known to be false in the case of 4 variables. Finally, I will discuss the possible generalization of the argument for $\mathsf{U}^k$ norms.