MAGIC087: Applied Algebraic Topology

Algebraic topology has been a key part of mathematics for at least a hundred years. Its main strength lies in its ability to detect shape of spaces. To provide qualitative description of spaces, for instance, how connected they are, various algebraic invariants have been developed, with simplicial cohomology being one of the most prominent. In recent years there has been a very strong impetus in developing direct applications of algebraic topology, which resulted in the creation of a new sub-field of applied algebraic topology. The subject is young, is developing very fast and offers numerous opportunities for research problems. The recent preoccupation with Big Data is probably the most important motivation, and it has led to rapid creation of new techniques capable of dealing with finite discrete spaces of huge complexity and dimensionality. This course will introduce the students to the key ideas of applied algebraic topology. Our main motivations will be applications to network theory and data analysis. This course is aimed at two kinds of students: pure mathematicians interested in applications, and applied mathematicians wishing to learn this new exciting theory straddling both disciplines. Additional reading will be suggested to support the course and to fill in any gaps in prerequisites. I will provide regular problem sheets. A special feature of the course will be additional exercises in Matlab. I will not assume any prior experience with Matlab, and the course will not depend on Matlab expertise, but it will be a good idea for those interested in applications to follow one of many online Matlab courses to learn the basics. Specific recommendations will be given. The course will introduce where appropriate specific examples of real-life data sets and networks; we will explain the problems, main difficulties, and solutions.

Basic topology and some algebraic topology, especially differential complexes and homology, cohomology. Some experience with Matlab would be useful, but guidance will be given to students who have not used this package before. An interest in crossing subject boundaries and a sense of intellectual adventure.

1. Topological spaces and topological invariants
2. Approximating topological spaces through simplicial complexes
3. Persistent cohomology and Betti numbers
4. Stability and periodic motion detection through persistent cohomology
5. Feature detection with persistent cohomology: barcodes.
6. Elementary ideas from simplicial Morse theory
7. Applications of Morse theory: mapper and similar tools.
8. Topology of sensor networks, coverage problems.
9. Topological data analysis
10. Shape classification and statistics