Introduction to topological data analysis 1000-1M24TDA

In this lecture, we will discuss the following topics:
1. Overview of topology: Mathematical and computational preliminaries. Metrics and
similarity measures. How topology appears, when we cannot precisely measure the
distances. What are the appropriate metrics for high dimensional data (to be cont. In
lect. 2)? Curse of dimensionality.
2. Broad overview of data analysis, difference between qualitative, descriptive, and
statistical methods. Focus on clustering methods. High dimensional data, dimension
reduction, and variable selection methods.
3. Approximating shapes: Complexes as examples of simple shapes that provide
combinatorial representations of complicated spaces. Examples: Simplicial
complexes, singular, point cloud-based, cubical, regular CW complexes, nerve
complexes etc. Introduction of efficient data structures to store those complexes.
4. Complexes from data: How to obtain complexes from trees, graphs etc (both
abstract and embedded). Relation to network theory, clustering coefficients and
similar concepts. Homotopy equivalence: homotopy equivalent to a sublevel set of
single or multi varied functions.
5. Topological equivalences and invariants: undecidability of homeomorphism
type, homotopy groups, computational limits.
6. Chains and cycles as generalization of paths and cycles in graphs (persistent)
homology and cohomology (in particular with Z2 coefficients), boundary matrix
reduction algorithm. Persistence diagrams, distances between them, further
requirements for statistics, need of vectorization.
7. Motivation for and limits to multi-parameter persistence, Euler characteristic
curves and profiles.
8. Topological Goodness-of-Fit tests in statistics.
9. Introduction to Discrete Morse Theory (DMT), connection of DMT and
filtrations / persistent homology. Iterated Morse complexes as a way to
compute (persistent) homology with field coefficients using Morse theory.
10. Reeb graphs, cover complexes, and mapper type algorithms, from the
point of view of plots of relations
(A, f (A))
for A being a subsample of a high dimensional set.
11. Standard mapper and Ball mapper, ClusterGraph.
12. Computational homotopy groups - relation to group representation. How
to get a simpler representation using DMT.
13. Geometrical estimators of Riemannian metrics on a manifold, estimators
of dimensionality, curvature, and reach from point clouds.
14. Dynamical systems, topology, Wazewski principle, Conley index.
15. Applications: Brain function, classification of neuron shapes, classification of
materials, lung structure in COPD, applications to economics and political sciences,
market prediction.