Seminar: Vector Bundles and Their Applications in Data Science

This seminar is about the interactions of vector bundles and data science. Vector bundles are a staple in the toolset of geometry and topology in mathematics. Their applications range from number theory to theoretical physics and data science. This seminar is about the theory of vector bundles and how it enters into the realms of data science.

The plan for the seminar is to first review the theory of vector bundles and then dive into some of their applications. We will proceed through the following topics (more precise assingment of talks will be discussed in the first meeting).

Part I : Theory

Manifolds and Lie Groups
Fiber and Vector Bundles
Principal Fiber Bundles and Associated Vector Bundles
Connections and Curvature for Vector Bundles
Characteristic Classes and Chern-Weil Theory
Classification theory of Vector Bundles

Part II : Applications

Vector Diffusion Maps
Heat Equations on Vector Bundles for Image Diffusion
Synchronization Problems and Fiber Bundle Theory

Assumed Background

Courses in analysis and linear algebra. Some prior exposure to differential geometry and topology is also assumed (scope: 1 term lecture on differential geometry or topology). The target audience for this seminar are master's students in mathematics, or computer science/scientific computing with a strong background in mathematics.

Guidelines

Language of the seminar will be English.
Present a 60 minute talk.
Your grade will only depend on your presentation, but we suggest that you share some form of notes for reference.
Active participation is strongly encouraged.
Either black board presentation or slideshow. We suggest to present most things on blackboard and only show results/visualization via slides.
In order to partake, you must sign up for the seminar in Heico, follow this link.

Details for the talks

We emphasize that this is mathematics seminar, meaning that you should also adhere to some form of rigorous notation. We also expect that you present at least one theorem during your talk. That is, your presentation should contain at least one formal mathematical statement whose significance you explain.

Furthermore, it is mandatory to arrange at least one meeting with us prior to your presentation, where we discuss what you are going to present during your talk. This meeting must be at least one week before your talk. Failing to meet this requirement will result in a failing grade.

Contact

If you are interested in participating in the seminar, feel free to contact us by mail for further information. See also the Müsli site for this seminar link.
Fabio Schlindwein schlindwein(at)math.uni(minus)heidelberg.de
Jonas Cassel: cassel(at)math.uni(minus)heidelberg.de

Prior Reading and Background Material

Prior reading (until first talk, 18.11)
Lie groups and algebras [Ch. 4 + 5.1].

Background material (for further reading)
Differential Geometry and Topology [1–3]
Bundle and Gauge Theory [4–6]
Characteristic Classes [4,7]

List of Talks

Introduction to Fiber and Vector Bundles (18.11) [6, Ch. 3+4]
fibre bundles, local trivializations, vector bundles, transition functions, structure group, group actions
Principal Fiber Bundles and Associated Vector Bundles (25.11) [6, Ch. 2+4] [5, Ch. 2.2-2.4]
Lie group representations principal fiber bundles, associated vector bundles
Connections and Curvature (02.12) [6, Ch. 5]
differential Forms, bundle-valued forms, Ehresmann connections, Curvature Two-Form, Structure Equation
Characteristic Classes and Chern-Weil Theory (09.12) [Reference]
de Rahm cohomology (long exact sequence), present the reference
Classification Theory (16.12)
present the reference [8] OR classification of flat bundles via holonomy [5, Ch. 4]
Application: Vector Diffusion Maps (13.01) [9]
discuss the paper
Further applications (20.01 + 27.01)
choose any of the following papers: image diffusion on vector bundles [10], synchronization [11], jigsaw puzzles [12], bundles in geometric deep learning [13–15]

References

1. Bredon GE (2013) Topology and geometry, Springer Science & Business Media.

2. Taubes CH (2011) Differential geometry: Bundles, connections, metrics and curvature, OUP Oxford.

3. Bott R, Tu LW (1982) Differential Forms in Algebraic Topology, New York, NY, Springer.

4. Husemöller D (1966) Fibre bundles, Springer.

5. Baum H (2009) Eichfeldtheorie, Springer.

6. Hamilton MJ (2017) Mathematical gauge theory, Springer.

7. Milnor JW, Stasheff JD (1974) Characteristic classes, Princeton university press.

8. Mitchell SA (https://sites.math.washington.edu/~mitchell/Notes/prin.pdf) Notes on principal bundles and classifying spaces.

9. Singer A, Wu H (2011) Vector Diffusion Maps and the Connection Laplacian.

10. Batard T (2011) Heat Equations on Vector Bundles—Application to Color Image Regularization. Journal of Mathematical Imaging and Vision 41: 59–85.

11. Gao T, Brodzki J, Mukherjee S (2019) The Geometry of Synchronization Problems and Learning Group Actions.

12. Huroyan V, Lerman G, Wu H-T (2020) Solving Jigsaw Puzzles by the Graph Connection Laplacian. SIAM Journal on Imaging Sciences 13: 1717–1753.

13. Gerken JE, Aronsson J, Carlsson O, et al. (2021) Geometric Deep Learning and Equivariant Neural Networks.

14. Bodnar C, Di Giovanni F, Chamberlain BP, et al. (2023) Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs.

15. Bamberger J, Barbero F, Dong X, et al. (2024) Bundle Neural Networks for message diffusion on graphs.