An inverse theorem for the gowers U^s+1[N]-norm

This is an announcement of the proof of the inverse conjecture for the Gowers U^s+1[N]-norm for all s ≥ 3; this is new for s ≥ 4, the cases s = 1, 2, 3 having been previously established. More precisely we outline a proof that if f: [N] → [-1, 1] is a function with kfkU^s+1[N] ≥ δ then there is a bounded-complexity s-step nilsequence F(g(n) Γ) which correlates with f, where the bounds on the complexity and correlation depend only on s and δ. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity. In particular, one obtains an asymptotic formula for the number of k-term arithmetic progressions p₁ < p₂ < · · · < p_k ≤ N of primes, for every k ≥ 3.