MORSE - Theoretical methods in Hamiltonian dynamics

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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.