I will give an overview on convenient calculus and differential geometry in infinite dimensions, with applications to diffeomorphism groups and shape spaces.
1. A short introduction to convenient calculus in infinite dimensions which is maily adapted to the the infinitely differentiable (smooth) case.
2. Manifolds of mappings (with compact source) and diffeomorphism groups as convenient manifolds.Spaces of planes curves will be treated with special emphasis.
3. A diagram of actions of diffeomorphism groups.
4. Riemannian geometries of spaces of immersions, diffeomorphism groups, and shape spaces, their geodesic equations with well posedness results and vanishing geodesic distance.
5. Riemannian geometries on spaces of Riemannian metrics and pulling them back to diffeomorphism groups.
6. Robust infinite dimensional Riemannian manifolds, and Riemannian homogeneous spaces of diffeomorphism groups.
We will discuss geodesic equations of many different metrics on these spaces and make contact to many well known equations (Cammassa-Holm, KdV, Hunter-Saxton, Euler for ideal fluids), if time permits.