Special Year Learning Seminar
Bounds in the Inverse Theorem for the Gowers Norms (for Certain Groups)
The inverse theorem for the Gowers U^{s+1}-norms has a central place in modern additive combinatorics, but all known proofs of it are difficult and most do not give effective bounds.
Over this seminar and the next, I will give an outline of a proof giving plausible bounds, in the case of groups Z/NZ where N is a large prime. Many of the key ideas are present in the cases s=2 and s=3, so I will focus on these for simplicity. The argument splits into an "algebraic part" which comprises a structure theorem for "local polynomials", and an "analytic part" which uses this to deduce the inverse theorem; the two may be taken somewhat separately.
I will assume some basic familiarity with the definition of the Gowers norms, and maybe some basic statements related to the inverse theorem, but not much else.