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

General information

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

 
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