General Prerequisites: Some familiarity with the main concepts from algebraic topology, homological algebra and category theory will be helpful.
Course Overview: Ideas and tools from algebraic topology have become more and more important in computational and applied areas of mathematics. This course will provide at the masters level an introduction to the main concepts of (co)homology theory, and explore areas of applications in data analysis.
Learning Outcomes: Students should gain a working knowledge of homology and cohomology computation for simplicial complexes and sheaves, and improve their geometric intuition. Furthermore, they should gain an awareness of a variety of applications (with an emphasis on data analysis).
Course Synopsis: This course consists of two parts:
1. the first part, comprising five weeks, will cover the basics of algebraic topology. Explicitly, the material includes
simplicial complexes, geometric realisations and simplicial maps; homotopy equivalence, carriers, nerves and fibres; homology and its computation; exact sequences and the snake lemma; cohomology, cup and cap products, poincare duality.
2. the second part consists of more advanced material. In the last three weeks, this course will examine persistent homology; cellular sheaves and their cohomology; discrete Morse theory.
