# Topological Methods in Data Analysis (dt. Topologische Methoden in der Datenanalyse)

 Level, degree of commitment Advanced module, compulsory elective module Forms of teaching and learning,workload Lecture (4 SWS), recitation class (2 SWS), 270 hours (90 h attendance, 180 h private study) Credit points,formal requirements 9 CP Course requirement(s): Successful completion of at least 50 percent of the points from the weekly exercises. Examination type: Written or oral examination (individual examination) Language,Grading English,The grading is done with 0 to 15 points according to the examination regulations for the degree program B.Sc. Mathematics. Duration,frequency One semester, irregular Person in charge of the module's outline Prof. Dr. Volkmar Welker

## Contents

Elementary theory of geometry and topology of simplicial complexes. Simplicial complexes and filtrations of simplicial complexes from metric data. Simplicial homology of simplicial complexes and persistent homology of filtrations. Algorithms for determining filtrations from metric data and algorithms for computing persistent homology. Interpretation of persistent homology in data analysis. Combination of persistent homology methods with classical methods.

## Qualification Goals

Students will

• Understand basic principles of elementary geometric and topological objects,
• penetrate algorithms that manipulate geometric and topological objects and compute their invariants,
• recognize invariants of geometric and topological objects as algebraic structures in other mathematical fields.
• have practiced mathematical ways of working (development of mathematical intuition and its formal justification, training of abstraction skills, reasoning),
• have improved oral communication skills in exercises through discussion and free speech in front of an audience.

## Prerequisites

None. The competences taught in the following modules are recommended: either Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis, Object-oriented Programming, Declarative Programming.

## Applicability

Module imported from B.Sc. Mathematics.

It can be attended at FB12 in study program(s)

• B.Sc. Mathematics
• M.Sc. Data Science
• M.Sc. Mathematics
• LAaG Mathematics

When studying M.Sc. Business Mathematics, this module can be attended in the study area Free Compulsory Elective Modules.

• H. Edelsbrunner. A Short Course in Computational Geometry and Topology. Springer, Heidelberg, Germany, 2014.
• H. Edelsbrunner and J. Harer. Computational Topology. An Introduction. Amer. Math. Soc., Providence, Rhode Island, 2010.

This page describes a module according to the latest valid module guide in Winter semester 2023/24. Most rules valid for a module are not covered by the examination regulations and can therefore be updated on a semesterly basis. The following versions are available in the online module guide:

• Winter 2016/17 (no corresponding element)
• Summer 2018 (no corresponding element)
• Winter 2018/19 (no corresponding element)
• Winter 2019/20 (no corresponding element)
• Winter 2020/21 (no corresponding element)
• Summer 2021 (no corresponding element)
• Winter 2021/22 (no corresponding element)
• Winter 2022/23 (no corresponding element)
• Winter 2023/24

The module guide contains all modules, independent of the current event offer. Please compare the current course catalogue in Marvin.

The information in this online module guide was created automatically. Legally binding is only the information in the examination regulations (Prüfungsordnung). If you notice any discrepancies or errors, we would be grateful for any advice.