Computational Topology

What it is about

You will learn the basics of algebraic topology with a focus on homology and its computation as well as concepts of persistent homology. In the end you will be able to compute the homology of simplicial complexes, work with coefficients, apply the fundamental exact sequences, and understand and sketch the key results of persistent homology.

What you will be able to do

• Understanding and application of elementary concepts of algebraic topology, especially homology
• Familiarity with simplicial complexes and simplicial maps
• Knowledge of fundamental statements about abelian groups
• Definitions: simplicial homology, induced maps, relative homology, homology with coefficients, cohomology
• Methods for computing homology with integer and field-based coefficients
• Understanding the exact homology sequence and its applications (e.g., Mayer–Vietoris, long exact sequence of a pair)
• Basic notions of categories and functors
• Duality of homology and cohomology over fields and their significance for persistent homology
• Ability to provide sketches of proofs for central results of persistent homology (e.g., stability of persistence barcodes, relation to extended persistence for PL-functions)

What the module consists of

• LectureConveying the theoretical content through examples and discussion
• Übung/Practice sessionsDeepening through exercises, weekly alternation between problem solving and presentations, submission and review of exercises

Teaching method

• Blackboard lecture with examplesIllustrative presentation of the content and discussion with the students
• Accompanying literature/handoutsLecture materials and further literature for self-study deepening
• Practice sheets and tutorialsApplication of methods, exercise control and feedback by the tutors