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Inverse theorem for the Gowers U(4) norm

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If you have a question about this talk, please contact Boris Bukh.

The Gowers norms, which come in a family U(2), U(3), U(4), ...are important objects of study in additive combinatorics. The inverse conjecture for the Gowers norms predicts that if the U(s+1) norm of a function f is at least delta then f correlates with an object called an s-step nilsequence. The resolution of this family of conjectures is the last uncompleted step in a programme of Tao and I to count quite general linear configurations of primes asymptotically.

I will review much of the above and then say something about the recent proof of the inverse conjecture for the U(4) norm by Tao, Ziegler and I. The U(2) result is classical and the U(3) inverse conjecture was established by Tao and I four years ago. It looks very likely that the new argument proves the general case and we are currently trying to write that up.

This talk is part of the Discrete Analysis Seminar series.

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