Sergey Zelik was one of two principle lecturers at a “crash course” on nonlinear PDEs held 8-9 November at the Centre for Analysis and Nonlinear PDEs at the Maxwell Institute in Edinburgh. Sergey gave 5 hours of lectures on “Invariant manifolds and finite-dimensional reduction for dissipative PDEs“.  A summary of the lectures is as follows: in these lectures, we analyze the possibility to reduce the long-time dynamics generated by a dissipative PDE to a finite-dimensional system of ODEs.  We start with the classical theory of inertial manifolds, then the elements of the Romanov theory which sometimes allow us to overcome the restrictive spectral gap conditions as well as the so-called spatial averaging method will be presented. Finally, the old and recent counterexamples which show the limitations of the theory as well as the necessity of the assumptions posed will be discussed. Website for CANPDE Lectures