"The goal of this project is to develop new techniques combining tools from dynamical systems, analysis and differential geometry to study the existence and properties of invariant manifolds arising from solutions to differential equations. These structures are relevant in the study of the qualitative properties of ODE and PDE and appear very naturally in important questions of mathematical physics. This proposal can be divided in three blocks: the study of periodic orbits and related dynamical structures of vector fields which are solutions to the Euler, Navier-Stokes or Magnetohydrodynamics equations (in the spirit of what is called topological fluid mechanics); the analysis of critical points and level sets of functions which are solutions to some elliptic or parabolic problems (e.g.eigenfunctions of the Laplacian or Green's functions); a very novel approach based on the nodal sets of a PDE to study the limit cycles of planar vector fields. With the introduction by the Principal Investigator, in collaboration with A. Enciso, of totally new techniques to prove the existence of solutions with prescribed invariant sets for a wide range of PDE, it is now possible to approach these apparently unrelated problems using the same strategy: the construction of local solutions with robust invariant sets and the subsequent uniform approximation by global solutions. Our recent proof of a well known conjecture in topological fluid mechanics, which was popularized by the works of Arnold and Moffatt in the 1960's, illustrates the power of this method. In this project, I intend to delve into and extend the pioneering techniques that we have developed to go significantly beyond the state of the art in some long-standing open problems on invariant manifolds posed by Ulam, Arnold and Yau, among others. This project will allow me to establish an internationally recognized research group in this area at the Instituto de Ciencias Matemáticas (ICMAT) in Madrid."