← Teaching

Geometry and Topology in Machine Learning

Master's course  /  Swiss Joint Master of Science in Computer Science  /  2026

Tuesdays 9:15–12:00  /  PER 21, UniFR, Room F207

This page is dedicated to the course Geometry and Topology in Machine Learning, which will be taught during the fall term of 2026 at the University of Fribourg. I will be uploading and updating the lecture materials as the course progresses in this webpage, so please check regularly!

Overview

The course will provide an overview of geometric and topological methods in machine learning, with a particular focus on deep learning, that is, learning methods based on neural networks. The goal of the course is to introduce some of the most well-known geometric and topological perspectives on data, such as graph theory, spectral methods, persistent homology, magnitude or discrete curvatures; and to explore their use within deep learning systems.

The introduction of geometry and topology in deep learning has typically taken one of two complementary approaches: either observational or interventional. The former seeks to leverage geometry and topology to better understand how deep learning systems behave, whereas the latter aims to incorporate geometry and topology within neural networks, either to allow them to learn from geometric and topological domains (graphs, simplicial complexes, cell complexes, hypergraphs, etc.) or to guide their learning injecting geometric or topological inductive biases. We will cover plenty of examples of both cases throughout the course.

Information about the sessions

This course will be taught in English, with a theoretical and a practical component. In the theory sessions, different geometric and topological tools for data analysis will be introduced, and their use in deep learning will be explored through examples drawn from current research. In the practical sessions, we will explore, through concrete examples again, how to implement these geometric and topological tools using Python and will cover problem sheets about the theory session. Please bring your laptops to the lectures if you want to run the notebooks on your own.

Assessment

Assessment will take the form of a collaborative project developed during the second half of the course, and an individual oral exam at the end of it.

Running the jupyter notebooks

In the code-based practice sessions, I will upload a jupyter notebook with sections for you to complete, and we will run through the notebook together in the session. I recommend running the notebooks in Google Colab, for minimal setup and because we have access to GPUs there. However, at the beginning of each notebook I will point out the dependencies of said notebook, in case you prefer to run them locally, so that you know what libraries you need. To run the notebooks in Google Colab press File ▸ Upload notebook, then Runtime ▸ Change runtime type ▸ GPU if you need access to GPUs.

Tentative calendar and syllabus

Lecture materials

Further references

Here I will be posting different references that you can use to complement the contents from the slides. Many are already mentioned there.

Lecture 1

There are many good references out there on deep learning these days. For instance, I can recommend the resources from the MIT 6.S191 course Introduction to Deep Learning and the series of videos by the YouTube channel 3Blue1Brown on Neural Networks.

Lecture 2

Again, graph theory is a wide subject that has been extensively studied, so there are plenty of references on the topic. For instance, you can take a look at:

• Grinberg, D. (2023). An introduction to graph theory. arXiv preprint, Chapter 2.

For a bit more of context on complexes and homology, check:

• R. Ghrist. (2014) Elementary Applied Topology, ed. 1.0, Createspace, Chapters 2 and 4. You can find the chapter pdfs at the author’s website.
• Hoppe, J., Grande, V. P., and Schaub, M. T. (2026). Don’t be Afraid of Cell Complexes! An Introduction from an Applied Perspective. arXiv preprint.
• Lim, L.-H. (2020). Hodge Laplacians on Graphs. SIAM Review, 62(3), 685–715. Link.

Lecture 3

• I recommend watching this talk by Petar Veličković, Theoretical Foundations of Graph Neural Networks, which inspired the first part of the lecture
• Also, take a look at the Geometric Deep Learning book, and multiple resources in the page.

Lecture 4

• For the maths behind the sections about partitioning graphs and the Cheeger inequalities, I really recommend chapters 20 and 21 from Spectral and Algebraic Graph Theory, by Daniel A. Spielman.
• To study over-squeasing, I recommend the paper On the Bottleneck of Graph Neural Networks and its Practical Implications, by Uri Alon and Eran Yahav, one of the first treatments of the topic in the literature.
• The following two papers concerning over-squashing, which are mentioned in the slides, are also very interesting but use geometric tools that we have not discussed in the course (and I am not sure we will have time to discuss them), but you can take a look at them as additional reading:
  • Understanding over-squashing and bottlenecks on graphs via curvature
  • Understanding Oversquashing in GNNs through the Lens of Effective Resistance

Lecture 5

For this lecture, I recommend this survey paper that reviews several dimension estimation methods, including the ones we studied in the lecture. Of course, you can also refer to the original research papers that we discussed in the lecture for the applications of these estimators to deep learning.

Contact the lecturers