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

在接下来的三年里，我打算了解混合物体，部分由构型空间和部分由重心空间和/或由临界点的不稳定流形集合组成的混合物体是如何在山部型问题和接触几何中出现的。我还想利用我为接触形式（在对偶Legendrian曲线空间上）定义的同调性作为一种实用工具。在一些科学领域（物理学：金斯堡-朗道，液晶，生物学，化学等），奇异解以一种自然的方式出现。光滑解是变分解，它们也以自然的方式出现，在相同的领域，但通过不同的过程（物理学中动能和势能的极值化……）我打算至少部分地了解连接这两个过程的桥梁是什么。

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.