Sergey Zelik gave a series of lectures at the Maxwell Institute in Edinburgh in 2011 and the lecture notes from that talk have now been published in the Proceedings of the Royal Society of Edinburgh.  The tittle of the paper is “Inertial manifolds and finite-dimensional reduction for dissipative PDEs“.  The notes are devoted to the problem of finite-dimensional reduction for parabolic partial differential equations.  Sergey gives a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mane Projection Theorems. The recent counter-examples showing that the underlying dynamics may in a sense be infinite dimensional if the spectral gap condition is violated, as well as a discussion of the most important open problems, are also included.  A link to the paper is here.