51±ΎΙ«

DAAD-DUHN

Topological Data Analysis: From Persistent Homology to Mapper

5–9 October 2026 Β· Β· 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 workflowβ€”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

  • I. Limonchenko & V. Chernyshev β€” Theoretical foundations: topology, homology and cohomology, persistent homology, barcodes and stability, toric topology.
  • L. Poliakova & O. Leonov β€” Practical track: data preprocessing, dimensionality reduction, the Mapper framework, TDA software, and topological model investigation.

Programme overview

  • Monday β€” Introduction: topology, persistent homology and first practical computations.
  • Tuesday β€” Persistent Homology, DR & Preprocessing: homology, data types, distance measures, dimensionality reduction and team formation.
  • Wednesday β€” Persistent Homology & Mapper Graph: persistent homology theory, the Mapper algorithm and guided campus activities.
  • Thursday β€” Advanced Topics & Mapper Application: toric topology, stability, Mapper implementation, quiz and project work.
  • Friday β€” Synthesis & Presentations: model investigation, research discussion and team presentations.

Monday, 5 October β€” Introduction

TimeFormatActivity / 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 β€” Persistent Homology, DR & Preprocessing

TimeFormatActivity / 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

Wednesday, 7 October β€” Persistent homology & Mapper Graph

TimeFormatActivity / 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 β€” Advanced Topics & Mapper Application

TimeFormatActivity / 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 β€” Synthesis & Presentations

TimeFormatActivity / 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 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.