Quantitative Inverse Theorem for Gowers Uniformity Norms 𝖴5 and 𝖴6 in 𝔽n2
In this talk, I will discuss a proof of a quantitative version of the inverse theorem for Gowers uniformity norms 𝖴5 and 𝖴6 in 𝔽n2. 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 Sym4 and Sym5 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 𝖴k norms.