09:00 to 09:30 Registration 09:35 to 09:45 Welcome from Christie Marr (INI Deputy Director) 09:45 to 10:30 Peter Michor (Universität Wien)General Sobolev metrics on the manifold of all Riemannian metrics Based on collaborations with M.Bauer, M.Bruveris, P.Harms. For a compact manifold $M^m$ equipped with a smooth fixed background Riemannian metric $\hat g$ we consider the space $\operatorname{Met}_{H^s(\hat g)}(M)$ of all Riemannian metrics of Sobolev class $H^s$ for real $s>\frac m2$ with respect to $\hat g$. The $L^2$-metric on $\operatorname{Met}_{C^\infty}(M)$ was considered by DeWitt, Ebin, Freed and Groisser, Gil-Medrano and Michor, Clarke. Sobolev metrics of integer order on $\operatorname{Met}_{C^\infty}(M)$ were considered in [M.Bauer, P.Harms, and P.W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. J. Differential Geom., 94(2):187-208, 2013.] In this talk we consider variants of these Sobolev metrics which include Sobolev metrics of any positive real (not integer) order $s$. We derive the geodesic equations and show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. INI 1 10:30 to 11:00 Morning Coffee 11:00 to 11:30 Jean Feydy (École Normale Supérieure); (ENS de Cachan)An efficient kernel product for automatic differentiation libraries, with applications to measure transport Authors : Benjamin Charlier, Jean Feydy, Joan Alexis Glaunès and Alain Trouvé This paper presents a memory-efficient implementation of the kernel matrix-vector product, which is suitable for use with automatic differentiation libraries -- in our case, PyTorch. This piece of software alleviates the major bottleneck of autodiff libraries as far as diffeomorphic image registration is concerned: symbolic python code can now scale up to large point clouds and shapes (100,000+ vertices). To showcase the value of automatic differentiation to the LDDMM community, we introduce the "normalized Hamiltonian" setting and show that it corresponds to a spatially regularized optimal transport of mass distributions: made tractable by autodiff libraries, the kernel normalization trick turns an extrinsic image deformation routine into an intrinsic measure transportation program. INI 1 11:30 to 12:15 tba INI 1 12:30 to 13:30 Lunch @ Wolfson Court 14:00 to 14:45 Klas Modin (Chalmers University of Technology)Riemannian Gradient Flows in Shape Analysis In this talk I show how the framework of Riemannian gradient flows on Lie group action orbits is connected to several branches of mathematics: optimal transport, information geometry, matrix decompositions, multivariate Gaussians, entropy flows, etc. The framework guides analysis, numerics, and software implementation. INI 1 14:45 to 15:30 Alain Goriely (University of Oxford)Morphoelasticity and the Geometry of Growth INI 1 15:30 to 16:00 Afternoon Tea 16:00 to 16:45 François Gay-Balmaz (CNRS - Ecole Normale Superieure Paris)Towards a geometric variational discretization of compressible fluid dynamics We present a geometric variational discretization of compressible fluid dynamics. The numerical scheme is obtained by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Our framework applies to irregular mesh discretizations in 2D and 3D. It systematically extends work previously made for incompressible fluids to the compressible case. We consider in detail the numerical scheme on 2D irregular simplicial meshes and evaluate the behavior of the scheme for the rotating shallow water equations. While our focus is fluid mechanics, our approach is potentially useful for discretizing problems involving evolution equations on diffeomorphism groups. This is a joint work with W. Bauer. INI 1 16:45 to 17:45 Welcome Wine Reception at INI