Non Compact Variational Problems in Contact Form Geometry and in Conformally Invariant Equations

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I intend over the next three years to understand how hybrid objects, made partly of configurations spaces and partly of bary-center spaces and/or of a collection of unstable manifolds of critical points arise in Yamabe-type problems as well as in contact form geometry.I also would like to make out of the homology which I have defined for contact forms (on a space of dual Legendrian curves) a practical tool.Singular solutions arise in a natural way in several fields of Science (Physics: Ginsburg - Landau, liquid crystals, biology, chemistry etc). Smooth solutions are variational solutions which also arise in a natural way, in the same fields, but through a different process (extremization of kinetic plus potential energy in Physics...) I intend to understand at least partly what are the bridges connecting these two processes.