Information on the Lecture: Geometric Data Science (WS 2020/2021)

General information

• Lecturer: JProf. Dr. Philipp Harms
  
• Assistant: Jakob Stiefel, M.Sc.
  
• Lecture: Wednesdays 10:15–11:45 on Zoom. Zoom password and lecture recordings on ILIAS. 
• Lecture notes: In collaboration with M. Bauer and P. W. Michor on ILIAS. 
• Exercises: Fridays 12:15-13:45 on Zoom. Zoom password and instructions on ILIAS.

Topic

Geometric data arise naturally in many scientific fields such as computational anatomy, brain connectivity, molecular biology, meteorology, oceanology, online navigation, social networks, and finance. Moreover, in everyday-life applications, depth-enhanced image data is produced by time-of-flight sensors in cars, game consoles, and recently also cell phone cameras. Analyzing such geometric data is a major challenge, as the configuration spaces of e.g. curves, surfaces, diffeomorphisms, graphs, etc. are infinite-dimensional nonlinear manifolds or more general stratified spaces. This course develops theoretical foundations for geometric data science, which are rooted in infinite-dimensional Riemannian geometry and combine methods of machine learning, statistics, and stochastics.

Announcements

• The first exercise sheet is due on November 11 at 10:15. 
• The first exercise class will be held on November 13 at 12:15.
• The last two exercise classes on February 5 and 12 will be held in a different format: we will collaboratively implement diffeomorphic landmark matching in the geomstats software library.
• For this reason, there is no exercise sheet to be solved for the last session.

Contents

Riemannian geometry as a basis for geometric data science and fluid dynamics

• Geometric data science
• Fluid dynamics
• Riemannian geometry on mapping spaces

Calculus beyond Banach spaces 

• Locally convex spaces
• Fréchet and Gâteaux differentiability
• Exponential law
• Convenient calculus
• Pitfalls

Manifolds of mappings

• Manifolds
• Fiber bundles
• Smooth mappings on compact domains
• Compactly supported smooth mappings
• Continuously differentiable mappings
• Sobolev mappings
• Shape spaces

Riemannian geometry in infinite dimensions

• Weak and strong Riemannian metrics
• Geodesic equation and Christoffel symbols
• Exponential map
• Geodesic distance
• Completeness and the Hopf–Rinow theorem
• Curvature
• Groups
• Quotients

Diffeomorphism groups

• Right-invariant Sobolev metrics
• Geodesic distance
• Geodesic equation
• Sobolev diffeomorphisms
• Well-posedness of the geodesic equation
• Diffeomorphic shape analysis

Embeddings and immersions

• Reparameterization-invariant Sobolev metrics
• Geodesic distance
• Geodesic equation
• Sobolev immersions
• Well-posedness of the geodesic equation
• Elastic shape analysis
• Varifold and current distances

Literature

Riemannian geometry as a basis for geometric data science and fluid dynamics

• B. Riemann. “Über die Hypothesen, welche der Geometrie zu Grunde liegen”. In: Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1868). Habilitationsschrift, 1854.
• D. W. Thompson. On Growth and Form. Reprint of 1942 2nd ed. (1st ed. 1917). Dover, 1992.
• V. Arnold. “Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits”. In: Ann. Inst. Fourier (Grenoble) 16.fasc. 1 (1966), pp. 319–361.

Locally convex spaces

• H. Jarchow. Locally convex spaces. Springer Science & Business Media, 2012.
• L. Narici and E. Beckenstein. Topological vector spaces. CRC Press, 2010.
• F. Treves. Topological Vector Spaces, Distributions and Kernels: Pure and Applied Mathematics, Vol. 25. Vol. 25. Elsevier, 2016.

Calculus beyond Banach spaces

• J. Boman. Differentiability of a function and of its compositions with functions of one variable. In: Math, Scand. 20 (1967), 249-268. 
• A. Frölicher and A. Kriegl. Linear spaces and differentiation theory. Pure and Applied Mathematics. Chichester: J. Wiley, 1988.
• H. H. Keller. Differential calculus in locally convex spaces. Springer Lecture Notes 417, 1974.
• A. Kriegl and P. W. Michor. The Convenient Setting for Global Analysis. ‘Surveys and Monographs 53’. Providence: AMS, 1997.
• A. Stacey. “Comparative smootheology”. In: Theory Appl. Categ. 25 (2011), No. 4, 64–117.

Manifolds of mappings

• K. Jänich. Vektoranalysis. Springer, 2005.
• S. Lang. Fundamentals of Differential Geometry. Springer, 1999.
• J. Lee. Introduction to Riemannian Manifolds. Springer, 2018.
• P. W. Michor. Topics in Differential Geometry. 'Graduate Studies in Mathematics 97'. Providence: AMS, 2008.

Riemannian geometry in infinite dimensions

•  R. Abraham, J. E. Marsden, and T. Ratiu. Manifolds, tensor analysis, and applications. 3rd ed. Vol. 75. Springer, 2012.
• W. P. A. Klingenberg. Riemannian Geometry. 2nd ed. Vol. 1. de Gruyter Studies in Mathematics. de Gruyter, 1995.
• S. Lang. Fundamentals of Differential Geometry. Vol. 191. Graduate Texts in Mathematics. Springer, 1999.

Diffeomorphism groups

• V. I. Arnold and B. A. Khesin. Topological Methods in Hydrodynamics. Vol. 125. Applied Mathematical Sciences. Springer, 1998.
• L. Younes. Shapes and Diffeomorphisms. Springer, 2010.

Embeddings and immersions

• M. Bauer, M. Bruveris, and P. W. Michor. Overview of the geometries of shape spaces and diffeomorphism groups. In: Journal of Mathematical Imaging and Vision 50 (2014), No. 1, 60-97.
• M. Bauer, N. Charon, P. Harms, H.-W. Hsieh. A numerical framework for elastic surface matching, comparison, and interpolation. arXiv:2006.11652