Name:  Alex Cohen  (MIT)

Title:    An Optimal inverse theorem for polynomials over large fields

Abstract: 

A polynomial f(x_1, … x_n) over a finite field has a large bias if its output distribution is far from uniform. It has rank `r' if we can write `f' as a function of polynomials g_1, …, g_r that each have smaller degree. Bias measures the amount of randomness, and rank measures the amount of structure. It is known that if `f' has small rank, it must have large bias. Green and Tao proved an inverse theorem stating that if `f' is significantly biased, its rank cannot be too large. Their bound was qualitative, however, and several authors gave quantitative improvements. We prove an optimal inverse theorem: the rank and the log of the bias are equivalent up to linear factors (over large enough fields). Our techniques are very different from the usual methods in this area, we rely on algebraic geometry rather than additive combinatorics. This is joint work with Guy Moshkovitz.