  We establish the existence of infinitely many \emph{polynomial} progressions
in the primes; more precisely, given any integer-valued polynomials $P_1, >...,
P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any
$\eps > 0$, we show that there are infinitely many integers $x,m$ with $1 \leq
m \leq x^\eps$ such that $x+P_1(m), ..., x+P_k(m)$ are simultaneously prime.
The arguments are based on those in Green and Tao, which treated the linear
case $P_i = (i-1)\m$ and $\eps=1$; 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.
