MORSE - Theoretical methods in Hamiltonian dynamics

自从Morse关于黎曼测地线的工作以来，Morse理论方法一直是哈密顿系统非微扰研究的有力工具。这些方法依赖于许多塑造哈密顿动力学的变分原理，其中最重要的是拉格朗日和哈密顿作用量。无限维Morse理论可以有效地研究前者的泛函，而后者需要更复杂的方法，如Floer理论，与许多优秀的数学理论一样，Floer理论在与其他理论相关时发挥了最大作用。对这种相互关系的探索还远未完成，对它们的更好理解有望帮助解决哈密顿动力学中的一些公开问题。Floer同调在周期轨道的研究中非常有效，但目前在同宿轨道的研究中只取得了有限的成功，其中经典变分方法仍然更有效。由于同宿和异宿是理解哈密顿系统整体行为的重要对象，因此需要更多的方法来解决这两个问题。我们的建议旨在解决这两个问题，并可概括为以下两个科学目标：1.发展哈密顿Floer理论的替代和补充方法，包括抽象泛函和特殊辛流形的环空间上的哈密顿作用泛函。改进现有的变分方法，开发新的工具来研究超越自然哈密顿系统类的哈密顿动力学中的同宿和异宿。

Since the work of Morse on Riemannian geodesics, Morse theoretical methods have been powerful tools in the non-perturbative study of Hamiltonian systems. These methods rely on many variational principles which shape Hamiltonian dynamics, the most important of which are the Lagrangian and the Hamiltonian actions. While the former functional can be efficiently studied by the infinite dimensional Morse theory, the latter one requires more sophisticated methods such as the Floer theory.As many good mathematical theories, Floer theory does its best when related to other ones. The quest for such interrelations is far from being completed, and a better understanding of them can be expected to help in solving some of the open problems in Hamiltonian dynamics.Floer homology is quite effective in the study of periodic orbits, but for now has had only a limited success in the study of homoclinic orbits, in which classical variational methods are still more efficient. Since homoclinics and heteroclinics are important objects in the understanding of global behaviour of Hamiltonian systems, it would be desirable to have more methods to tackle them.Our proposal intends to address these two issues and can be summarised into the following two scientific aims:1. Develop alternative and complementary approaches to Hamiltonian Floer theory, both for abstract functionals and for the Hamiltonian action functional on the loop space of particular symplectic manifolds.2. Refine the existing variational methods and develop new tools for the study of homoclinics and heteroclinics in Hamiltonian dynamics beyond the class of natural Hamiltonian systems.