OneMath World School on Topological Data Analysis

20–28 February, 2026

About

This winter school is part of the OneMath World School, a global effort coordinating several mathematics institutes to organise schools on emerging topics in mathematics. It is a collaboration among several institutes.

• Casa Matemática Oaxaca (CMO), Mexico
• Chennai Mathematical Institute (CMI), India
• Institute of Mathematics of Granada (IMAG), Spain
• Brin Mathematics Research Center (BMRC), USA
• Intelligent Analytics (IA) at UBC-Okanagan, Canada
• Banff International Research Station (BIRS), Canada

Academic details

This workshop aims to provide a practical introduction to topological data analysis (TDA) with a focus on hands-on techniques and implementation. Participants will learn how to apply TDA methods, particularly persistent homology and the Mapper algorithm, to extract meaningful features from complex datasets. There will be emphasis on applications and use of Python libraries for implementation. Algebraic topology will be used as a tool, with a focus on understanding the outputs and their applications rather than the underlying theory.

Boot Camp: 20–21 February

A boot camp will be organized before the main workshop, covering the following topics.

• A crash course on simplicial complexes and related combinatorial techniques to model topological problems.

• Simplicial homology and related linear algebra.

• A crash course in machine learning with a particular emphasis on classification algorithms and kernels.

• Tutorials to help with ML using Python.

Mini-course on persistent homology and topological machine learning: 23–26 February

There will be lectures in the morning and tutorial sessions in the afternoon. The morning lectures will start from scratch and gradually get into the intricacies of topological data analysis. The afternoon tutorials will impart hands-on training.

• Lecture series 1 - Introduction to Persistent Homology

  1. Filtrations of topological spaces.

  2. Constructing simplicial complexes from point-cloud data.

  3. Persistent barcodes, diagrams and their visualization.

  4. Introduction to the scikit-tda library.

  5. Distances and stability.

• Lecture series 2 - Topological machine learning

  1. Statistics of persistence diagrams and landscapes.

  2. Persistence images: converting persistence diagrams into vectors.

  3. Persistence entropy: quantifying the complexity of topological features.

  4. Other descriptors: Betti curves, silhouette functions, kernel methods (overview).

  5. Topological feature engineering for classification and regression.

  6. Unsupervised learning with topological descriptors.

  7. Time series analysis using persistent homology.

• Lecture series 3 - The mapper algorithm

  1. Morse theory, Reeb graphs and their properties.

  2. Topological mapper.

  3. Building the Mapper graph: nodes, edges, and their interpretation.

  4. Parameter selection: cover, overlap, filter function.

  5. Introduction to the KeplerMapper library in Python.

  6. Feature selection and data summarization.

  7. Comparative study of Mapper and other visualization techniques.

• In addition, there will be two overview lectures.

  1. Techniques other than persistent homology and Morse theory.

  2. Geometric deep learning.

Applications of TDA: 27–28 February

The idea is to identify key papers in the following areas and assign them to participants. They will work in groups, in these 2 days they will understand the setup and try to recreate the experiments. By the end they would have identified some open problems to work on in the future.

• Various aspects of image analysis: boundary detection, classification, processing, retrieval, segmentation etc.

• Applications to time series and signal processing via Taken's embedding theorem and other techniques from dynamical systems.

• Robotics.

Speakers

To be announced.

Participation

• Proposed audience: Masters, Ph.D. students, postdocs, and early career researchers in the areas of Mathematics and Computer Science.

• Numbers: Up to 40 participants.

• Applying for participation: Procedure to be announced.

Schedule

• Dates: 20–28 February, 2026
• Boot camp: 20–21 February, 2026
• Main workshop: 23–28 February, 2026