Lie Theory and Applications. Iii

Geometry and analysis of shape space. The core of shape space in one of its simplest forms is the orbit space of the action of the group of diffeomorphisms of the circle S1 (the reparametrization group) on the space of immersions of the circle into the plane R2. The aim is to find good Riemannian metrics which allow applications in pattern recognition and visualization. The contributions of the PI were obtained mainly in collaboration with David Mumford. In the paper [M107] via the Hamiltonian approach many metrics were investigated, together with their conserved quantities (one of them is the reparameterization momentum) and their sectional curvatures. Recall from a result of the predecessor of this project, that the L2-metric on the space of immersions has zero geodesic distance on the orbit space under the reparameterization group. These metrics come in 3 flavors: Some are derived from the the L2-metric by multiplying it with a function of the length of the curve (a conformal change) or by multiplying the L2-integrand with a function of length and curvature (this is called almost local). Of the second flavor are the metrics which come from the Sobolev Hn-inner product on the space of immersions. The last flavor comes from using a suitable Sobolov metric on the diffeomorphism group of the plane and treating shape space as a homogeneous space. This is the metric that has found already many applications, particularly in medicine. This metric was pioneered by Michael Miller, Alain Trouve, and Laurent Younes, see [3, 4, 9] These applications are done at the ‘Center for imaging sciences’ at the Johns Hopkins University, and they are used to recognize diseases from the the tomographical data of hearts and brains, [6, 7]. A version of one of the metrics in [M107] turns out to be isometric to a quotient of (an open subset in) an infinite dimensional Grassmannian of 2-planes in Hilbert space. Here recent