In this course, we will tackle the following. 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.

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.