We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution W which gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev space H¹ than the one of W, then we have global well-posedness and scattering. If the norm is larger than the one of W, then we have break-down in finite time.

We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P₁, …, P_k ∈ Z[m] in one unknown m with P₁(0) = … = P_k(0) = 0, and given any ε > 0, we show that there are infinitely many integers x and m, with $1 \leqslant m \leqslant x^\varepsilon$, such that x + P₁(m), …, x + P_k(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case P_j = (j − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties.