# Project B8: Symplectic Geometry - Theory and Applications to Dynamics

## Principal investigators

## Participating scientists

## Summary

This project aims at developing theoretical aspects of Floer and holomorphic curve theory further, in particular using polyfold theory and applying them to dynamical systems. These will encompass certain classical dynamical systems such as the restricted three body problem (3BP) but also dynamical applications extending our previous work on Rabinowitz Floer theory.
Basically all Floer theories suffer from transversality problems unless they are considered on special symplectic or contact manifolds. In order to overcome this, Hofer-Wysocki-Zehnder developed the fundamentally new Fredholm theory in polyfolds. Incidentally to compute various Floer theories usually symmetries are employed thus adding to the transversality problems. One goal is therefore to establish a general abstract perturbation result in polyfolds with symmetry. As an initial application we will study a model problem in Morse-Bott theory with the ultimate goal of replacing Morse by Floer theory.
Only very recently a study of the planar restricted 3BP with modern tools from symplectic geometry has been initiated. We are now in the position to apply foliation techniques by holomorphic curves to the planar restricted 3BP. The planar restricted 3BP is a very complicated dynamical system. In order to get a better understanding we plan on examining some approximate problems such as the rotating Kepler, Euler's and Hill's lunar problem. At the same time we will try to address some easier questions concerning the planar restricted 3BP using holomorphic curve theory.