- I. Limonchenko & V. Chernyshev — Theoretical foundations: topology, homology and cohomology, persistent homology, barcodes and stability, toric topology.
- L. Poliakova & O. Leonov — Practical track: data preprocessing, dimensionality reduction, the Mapper framework, TDA software, and topological model investigation.
Topological Data Analysis: From Persistent Homology to Mapper
5–9 October 2026 · · Ulm University
The school flyer is available for download.
About the school
Learn how topology reveals structure in complex data. Through lectures, coding sessions and a team project, participants will build an end-to-end topological data analysis workflow—from data preprocessing and persistent homology to Mapper graphs and their interpretation.
The programme combines rigorous mathematical foundations with practical data analysis. Particular attention is given to the Mapper framework, a flexible approach for visualising and investigating the shape of complex, high-dimensional datasets.
At a glance
- An intensive five-day programme (5–9 October 2026) for Master's students, PhD students and PostDocs in Mathematics and Computer Science.
- 4 expert speakers covering both theory and practice.
- Python-based sessions combining lectures with coding.
- Team project with a final presentation.
What you will learn
- Understand the topological foundations of modern topological data analysis (TDA).
- Build simplicial complexes and compute persistent homology.
- Select suitable distance measures and preprocessing methods for different data types.
- Compare PCA (Principal Component Analysis), t-SNE (t-distributed Stochastic Neighbor Embedding), UMAP (Uniform Manifold Approximation and Projection), Isomap (Isometric Mapping), and MDS (Multidimensional Scaling) in practical workflows.
- Construct, tune, and interpret Mapper graphs.
- Apply TDA tools to a team project and present a reproducible analysis.
Programme highlights
Mathematical foundations
Topology, homology and cohomology, persistent homology, barcodes and stability.
Hands-on practice
Data preprocessing, metrics, dimensionality reduction, clustering, and TDA software.
Mapper framework
Filters, covers, clustering, nerve construction, parameter tuning, and model interpretation.
Team project
Small teams select a dataset, develop an analysis, and present their findings.
Academic exchange
Personal talks, guided discussions, campus activities, and networking.
Lecturers
Programme overview
- Monday — Introduction: topology, persistent homology and first practical computations.
- Tuesday — Persistent Homology, DR & Preprocessing: homology, data types, distance measures, dimensionality reduction and team formation.
- Wednesday — Persistent Homology & Mapper Graph: persistent homology theory, the Mapper algorithm and guided campus activities.
- Thursday — Advanced Topics & Mapper Application: toric topology, stability, Mapper implementation, quiz and project work.
- Friday — Synthesis & Presentations: model investigation, research discussion and team presentations.
Monday, 5 October — Introduction
| Time | Format | Activity / topic |
| 09:00–10:20 | Lecture | Introduction to Topology |
| 10:20–10:50 | Break | Coffee break |
| 10:50–12:10 | Lecture | From geometry to topology: a gentle introduction to persistent homology |
| 12:10–13:40 | Meal | Lunch |
| 13:40–15:00 | Practical | Building simplicial complexes and computing persistence with GUDHI, Ripser and giotto-tda |
| 15:00–15:30 | Break | Coffee break |
| 15:30–17:30 | Welcome | Welcome reception and personal talks |
Tuesday, 6 October — Persistent Homology, DR & Preprocessing
| Time | Format | Activity / topic |
| 09:00–10:20 | Lecture | Homology and Cohomology |
| 10:20–10:50 | Break | Coffee break |
| 10:50–12:10 | Practical | Data types, features and targets; selecting distance and similarity measures |
| 12:10–13:40 | Meal | Lunch |
| 13:40–15:00 | Practical | Dimensionality reduction: PCA, t-SNE, UMAP, Isomap and MDS |
| 15:00–15:30 | Break | Coffee break |
| 15:30–16:00 | Quiz | Formative, non-graded recap of Days 1–2 |
| 16:00–17:30 | Project | Team formation and dataset brainstorming |
| 17:45 | Dinner |
Wednesday, 7 October — Persistent homology & Mapper Graph
| Time | Format | Activity / topic |
| 09:00–10:20 | Lecture | Mapper algorithm: filters, covers, clustering, and nerve construction |
| 10:20–10:50 | Break | Coffee break |
| 10:50–12:10 | Lecture | Persistent homology theory II |
| 12:10–13:40 | Meal | Lunch |
| 13:40–15:40 | Tour | Campus tour |
| 15:40–18:00 | Break | Free time |
| 18:00 | Tour | City tour |
Thursday, 8 October — Advanced Topics & Mapper Application
| Time | Format | Activity / topic |
| 09:00–10:20 | Lecture | Toric Topology and TDA |
| 10:20–10:50 | Break | Coffee break |
| 10:50–12:10 | Lecture | Persistent homology theory III: barcodes and stability |
| 12:10–13:40 | Meal | Lunch |
| 13:40–15:00 | Practical | Building Mapper graphs with Kepler Mapper and giotto-tda; parameter tuning |
| 15:00–15:30 | Break | Coffee break |
| 15:30–16:00 | Quiz | Formative, non-graded recap of Days 3–4 |
| 16:00–17:30 | Project | Team project work and lecturer consultations |
Friday, 9 October — Synthesis & Presentations
| Time | Format | Activity / topic |
| 09:00–10:20 | Lecture | Investigating topological models |
| 10:20–10:50 | Break | Coffee break |
| 10:50–12:10 | Discussion | Q&A, open problems, and research directions |
| 12:10–13:40 | Meal | Lunch |
| 13:40–15:40 | Presentations | Team presentations and closing ceremony |
| 15:40–16:10 | Break | Coffee break |
| From 16:10 | Departure | Participant departure |
Registration
Registration is now open. Applications have to be submitted by 31 August 2026. To apply, please email the organisers and attach a motivation letter, a short CV, and a letter of recommendation from your academic supervisor.
Applicants will be selected on a competitive basis, taking into account their academic background, motivation, and the relevance of the school to their studies.
There is no registration fee, but participants are expected to cover their own travel and accommodation expenses. A limited amount of financial support for travel and accommodation is available to Ukrainian participants on a competitive basis. If you require financial support, please indicate this in your application.
Who can apply
Applications are welcome from Master’s students, PhD students, and early-career researchers (postdoctoral researchers) in mathematics, computer science, and closely related fields. Applicants should have a solid foundation in mathematics at Bachelor’s level. Basic experience with Python is recommended. Previous coursework in topology is not required.
Cancellation and contact
If you can no longer attend, please inform the organising team as soon as possible so that the place can be offered to another applicant.
Venue
The school will take place at the at Ulm University. The modern seminar building provides flexible teaching spaces for lectures, coding sessions, and teamwork.
Arrival information
Detailed arrival instructions, campus maps, and meeting points for the tours will be sent to accepted participants before the school.
Preparation
Participants are expected to bring a laptop for the hands-on sessions.
Frequently asked questions
No.
No.
No. The programme starts with a guided introduction. A solid general mathematics background is expected.
Basic Python experience is recommended.
Yes. A personal laptop is required for the hands-on sessions.
English.
Yes. Participants are encouraged to suggest datasets and project topics.
No. The quizzes are formative and intended to support learning.
Organisers and support
The school is organised by Dr. Mikhail Chebunin and Prof. Dr. Evgeny Spodarev within the DUHN–DAAD-supported Double-Degree Programme Ulm–Kharkiv. It aims to strengthen academic exchange between the two universities and to provide students with advanced training in modern mathematical data analysis.
