  We prove a center manifold theorem for rough partial differential equations
(rough PDEs). The class of rough PDEs we consider contains as a key subclass
reaction-diffusion equations driven by nonlinear multiplicative noise, where
the stochastic forcing is given by a $\gamma$-H\"older rough path, for
$\gamma\in(1/3,1/2]$. Our proof technique relies upon the theory of rough paths
and analytic semigroups in combination with a discretized Lyapunov-Perron-type
method in a suitable scale of interpolation spaces. The resulting center
manifold is a random manifold in the sense of the theory of random dynamical
systems (RDS). We also illustrate our main theorem for reaction-diffusion
equations as well as for the Swift-Hohenberg equation.
