ETEX01 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 students 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 by topological complexes and filtrations and, calculate the persistent homology of these mathematical objects and the persistent diagrams. After the course the participants should be able to give reliable interpretations of the topological invariants in applications in technology and humanities. Besides, they should be able to use software packages to calculate homology of complexes and persistent diagrams, and to carry out analysis data with these methods.

Contents:

• Homology: CW-complexes and simplicial complexes. Construction of complexes from data: Cech and Rips complexes. Complex homology. Methods to calculate homology



• Persistent Homology: Definition of persistent homology, persistent diagrams. Calculations of persistent diagrams. Filtrations and Discrete Morse Theory. Stability Theorem.



• TDA and Other Applications: What is TDA, examples and applications. Applications to discrete optimization. Construction of complexes from data. Digital topology. Computer Vision.

Teaching and Organization:

The course is a hybrid course, with an intensive part of two weeks lectures on campus, and a virtual part for the rest of the Fall Term. One can find the schedule of the lectures below.

The lectures will be broadcasted on Zoom and recorded.

Materials like lecture notes, lecture videos, collection of examples, training exercises - to be solved!!, algorithms, etc will be posted on the virtual aula of the course at LISAM.

The virtual aula has also a chat/forum function. Use it.

Examination

The examination consists of doing a project. The project can be more theoretical, like studying some research articles, or more applied and computational like working out a large concrete example for some data set.

The projects can be carried out in groups of two or three students

Examiner

Milagros Izquierdo

Schedule

Timeedit

Page responsible: milagros.izquierdo@liu.se
Last updated: 2024-09-01