MA341 – Applied and Computational Topology, Fall 2025

• Assignment 1 due Friday, Oct. 24


• A sample of applications of knots and links in physics:
  • Zi Qiang Qiu, Topology in magnetic systems (including linking number as defined by Gauss), 2018.
  • Kai Wang, Avik Dutt, Charles C. Wojcik, Shanhui Fan, Topological complex-energy braiding of non-Hermitian bands, 2021.


• Zhen Zhang, Examples of graph data, 2020. See [CVJ, Section 2.5] for more examples. You are also encouraged to read about graph neural networks.


• A classic and prototypical series of applications of TDA with image and video data: distributions and machine learning
  • Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, Afra Zomorodian, On the local behavior of spaces of natural images, 2008.
  • Rickard Brüel Gabrielsson, Gunnar Carlsson, Exposition and interpretation of the topology of neural networks, 2019.
  • Ephy R. Love, Benjamin Filippenko, Vasileios Maroulas, Gunnar Carlsson, Topological convolutional layers for deep learning, 2023.

Instructor

朱一飞 ZHU Yifei

CoS-M705

zhuyf@sustech.edu.cn

Office hours: Thursdays 2:00–3:50 pm or by appointment

TA: 张海宇 ZHANG Haiyu

Prerequisites

Topology (MA323)

Objectives

Applied and computational topology has become a subject that applies to a wide range of topics. This includes pattern recognition in data science, and genomics and evolution in biology, notably through the method of persistent homology. It also employs computer softwares to study questions internal to topology and geometry. We aim to gain an overview of the subject, learn its basic theory and examples, with an emphasis on persistent homology and its applications.

References

• [EH] Herbert Edelsbrunner and John L. Harer, Computational topology: An introduction, American Mathematical Society, 2010.

• [CVJ] Gunnar Carlsson and Mikael Vejdemo-Johansson, Topological data analysis with applications, Cambridge University Press, 2022.
• Raúl Rabadán and Andrew J. Blumberg, Topological data analysis for genomics and evolution: Topology in biology, Cambridge University Press, 2020.

• [DONUT] Barbara Giunti, Jānis Lazovskis, and Bastian Rieck, DONUT: Database of Original & Non-Theoretical Uses of Topology, 2022.
• Kelsey Houston-Edwards, How squishy math is revealing doughnuts in the brain, Scientific American, 2022.
• Bastian Rieck, Topology meets machine learning: An introduction using the Euler characteristic transform, Notices of the American Mathematical Society, 2025.
• This course grew out of a reading seminar on the subject in Fall 2020 and Spring 2021.

Exams

There will be none. Instead, by the end of the semester, you are required to submit a TeX'd report on a specific real-world application of computational topology, based on reading the literature (see above, esp. [DONUT] and [CVJ, Chapter 6]) and reproducing some of the results therein, or even better, carrying out analogous experiments of your own. This is worth 40% of your final grade. More specific instructions will be provided later in the course. Meanwhile, you are always welcome to discuss your plans and specific questions related to this with your instructor and TA.

Homework

There will be about 5 assignments, every 2 or 3 weeks, listed at the top of this page. Homework is worth 60% of your final grade. You must make arrangements in advance if you will not be handing in homework on time. We encourage you to discuss homework problems with your classmates, including strategies for solving different kinds of problems. However, when you actually write up your solutions, you must do this on your own.