Applied Topology, TDA

Aim

Topological Data Analysis has become an important tool in machine learning and artificial intelligence. The course will provide the needed concepts, methods and tools in homology (persistent homology and persistent diagrams) to have efficient methods to study data clouds and other objects in data analysis.

After the course the student should be acquainted with the basic concepts in topological complexes, homology and persistent homology, and its use in data analysis and other subjects. Specifically, they should be able to:

• approximate data clouds and other geometrical objects bytopological complexes and filtrations and, calculate the persistent homology of these mathematical objects and the persistent diagrams.
• give reliable interpretations of the topological invariants in applications in technology and humanities.
• use software packages to calculate homology of complexes andpersistent diagrams, and to carry out analysis data with these methods.

Content

• Homology: CW-complexes and simplicial complexes. Construction of complexes from data: Čech and Rips complexes. Filtrations and Discrete Morse Theory. Complex homology. Methods to calculate homology.
• Persistent Homology: Definition of persistent homology, persistent diagrams. Calculations of persistent diagrams. Stability Theorem.
• TDA and Other Applications: What is TDA, examples and applications. Applications to discrete optimization. Construction of complexes from data. Digital topology. Spaces of configurations.