Tom Mestdag's part of the course starts with
(1) Elementary calculus of variations, and its applications.
(2) Lie groups, actions of Lie groups on differentiable manifolds.
(3) Fibre bundles (vector bundles, principal fibre bundles, etc), connections on fibre bundles.
Then the course usually focuses on
(a) Vector fields with a symmetry group, reduction, reconstruction.
(b) Applications in the context of Lagrangian systems, of symplectic geometry and of Poisson geometry.
Sonja Hohloch's part of the course studies properties of integrable Hamiltonian PDEs with the Korteweg-de Vries (KdV) equation as leading example, i.e., several items of Tom Mestdag's part reappear in an infinite dimensional setting. More precisely, we study (at least) the following topics concerning the KdV equation:
(1) Motivation from physics via soliton waves and intuition how to get from the wave equstion to KdV.
(2) Hamiltonian and integrable properties of KdV: the KdV hierarchy and its motivation.
(3) Existence and uniqueness questions of solutions of KdV.
(4) Symmetries of KdV.
(5) Scattering and inverse scattering of KdV.