Overview on Convenient Calculus and Differential Geometry in Infinite dimensions, with Applications to Diffeomorphism Groups and Shape Spaces

Peter Michor

(Faculty of Mathematics, University of Vienna)

Event: ERC School on Geometric Evolution Problems

Date: Jun 23, 2014, time: 09:30

Abstract. 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

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.

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