  In this note we announce 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 (details
of which will appear in a forthcoming paper) that if f : [N] -> [-1,1] is a
function with || f ||_{U^{s+1}[N]} => \delta then there is a bounded-complexity
s-step nilsequence F(g(n)\Gamma) which correlates with f, where the bounds on
the complexity and correlation depend only on s and \delta. 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 p_1 < p_2 <
... < p_k <= N of primes, for every k => 3.
