This is a 9 ECTS (270 h) course targeted at master-level Computer Science and Mathematics students.

Fundamental concepts of relative homology and cohomology theory and persistence theory in computational settings, category theory in this context, algorithms for the computation of (persistent) homology, (extended) persistence modules and their decompositions, Morse theory, duality theorems, quiver representation theory, stability of persistence diagrams and barcodes, algebraic stability, topological filtrations, multiparameter persistence, invariants of persistence, topological data analysis, applications to shape pattern recognition, machine learning, identification of geometric objects.

While having knowledge of homology and other methods of algebraic topology is certainly helpful, the goal of the course is to be largely self-contained. Basic knowledge of linear algebra, algorithms, data structures, and complexity analysis are assumed.

There will be oral exams at the end of the semester.

The lectures will not adhere to any book. Instead, lecture notes made specifically for the course will be made available. Helpful literature for the course includes the following.

- Herbert Edelsbrunner, John Harer (2010). Computational Topology: An Introduction. American Mathematical Society.

- Steve Oudot (2015). Persistence Theory: From Quiver Representations to Data Analysis (Vol. 209). American Mathematical Society.

- Magnus Bakke Botnan, Michael Lesnick (2022). An Introduction to Multiparameter Persistence. https://arxiv.org/abs/2203.14289.

- Allen Hatcher (2002). Algebraic Topology (Vol. 44). Cambridge University Press.