Title: An optimal inverse theorem
Abstract: The geometric rank of a k-tensor, or a (k-1)-linear map, is the codimension of its kernel variety, which is the variety cut out by the (k-1)-linear forms (for k=2 this is simply matrix rank).
Using a carefully chosen subvariety of the kernel that satisfies certain smoothness and F-rationality properties, together with a new iterative process for decomposing successive derivatives of a tensor on a variety, we prove that the partition rank of Naslund and the analytic rank of Gowers and Wolf are equivalent, up to a constant depending on k, over any large enough finite field. Proving the equivalence between these two quantities is the main question in the "bias implies low rank" line of work in higher-order Fourier analysis, and was reiterated by multiple authors.
Joint work with Alex Cohen.