- 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 · Dr. Barbara Mez-Starck-Haus · Ulm University
About the school
The school showed how topology reveals structure in complex data. Through lectures, coding sessions and a team project, participants built an end-to-end topological data analysis workflow—from data preprocessing and persistent homology to Mapper graphs and their interpretation.
The programme combined rigorous mathematical foundations with practical data analysis. Particular attention was given to the Mapper framework, a flexible approach for visualising and investigating the shape of complex, high-dimensional datasets.
The school flyer and short program are available for download.
Read the report on the school.
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 covered both theory and practice.
- Python-based sessions combined lectures with coding.
- Team project with a final presentation.
What participants learned
- The topological foundations of modern topological data analysis (TDA).
- Building simplicial complexes and computing persistent homology.
- Selecting suitable distance measures and preprocessing methods for different data types.
- Comparing 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.
- Constructing, tuning and interpreting Mapper graphs.
- Applying TDA tools in a team project and presenting 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 selected a dataset, developed an analysis and presented 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 (Limonchenko) |
| 10:20–10:50 | Break | Coffee break |
| 10:50–12:10 | Lecture | From geometry to topology: a gentle introduction to persistent homology (Chernyshev) |
| 12:10–13:40 | Meal | Lunch |
| 13:40–15:00 | Practical | Building simplicial complexes and computing persistence with GUDHI, Ripser and giotto-tda (Poliakova) |
| 15:00–15:30 | Break | Coffee break |
| 15:30–15:45 | Introduction | Opening / Preparation for contributed talks |
| 15:45–16:00 | Contributed Talk |
Serhii Zhuravkov Beyond the Single Label: A Concept-Lattice and Persistent Homology Perspective on Job Matching |
| 16:00–16:15 | Contributed Talk |
Kateryna Chekhovskaia Topological vectorization methods, their integration with machine learning and deep learning architectures |
| 16:15–16:30 | Break | Short break |
| 16:30–17:30 | Welcome | Welcome reception |
Tuesday, 6 October — Persistent Homology, DR & Preprocessing
| Time | Format | Activity / topic |
| 09:00–10:20 | Lecture | Homology and Cohomology (Limonchenko) |
| 10:20–10:50 | Break | Coffee break |
| 10:50–12:10 | Practical | Data types, features and targets; selecting distance and similarity measures (Leonov) |
| 12:10–13:40 | Meal | Lunch |
| 13:40–15:00 | Practical | Dimensionality reduction: PCA, t-SNE, UMAP, Isomap and MDS (Poliakova) |
| 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 |
| 18:30 | Dinner | Сonference 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 (Leonov) |
| 10:20–10:50 | Break | Coffee break |
| 10:50–12:10 | Lecture | Persistent homology theory II (Chernyshev) |
| 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 (Limonchenko) |
| 10:20–10:50 | Break | Coffee break |
| 10:50–12:10 | Lecture | Persistent homology theory III: barcodes and stability (Chernyshev) |
| 12:10–13:40 | Meal | Lunch |
| 13:40–15:00 | Practical | Building Mapper graphs with Kepler Mapper and giotto-tda; parameter tuning (Poliakova) |
| 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 (Leonov) |
| 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 |
Venue
The school took place at the Dr. Barbara Mez-Starck-Haus at Ulm University. The modern seminar building provided flexible teaching spaces for lectures, coding sessions, and teamwork.
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 was organised by Dr. Mikhail Chebunin and Prof. Dr. Evgeny Spodarev within the DUHN–DAAD-supported Double-Degree Programme Ulm–Kharkiv. It aimed to strengthen academic exchange between the two universities and to provide students with advanced training in modern mathematical data analysis.
