Introduction to topological data analysis

General information

This is the website for my introductory lecture on topological data analysis, held at Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw. The lecture takes place on Tuesdays 16:15-18:00 in room 4050. The exercise classes take place on Tuesdays 18:15-20:00, also in room 4050. You can find the lecture on USOS here.

Lecture slides and notes

1. General overview on topology

2. Overview on Data Analysis with a focus on statistical methods, clustering, and dimension reduction

3. Cell complexes, cubical complexes, regular CW complexes, and simplicial complexes

4. The simplex tree and baricentric subdivisions ; Complexes from data points

5. Simplicial homology and persistent homology ; additional slides

6. Decomposition of persistence modules and statement of the stability theorem with respect to Hausdorff distance

7. Proof of the stability theorem with respect to Hausdorff distance

8. Multiparameter persistence and Euler characteristic

9. Goodness of fit tests

10. Reeb Graph and Mapper algorithm

11. Ball Mapper and Cluster Graph

12. Discrete Morse Theory (slides with example)

13. Iterated Morse complexes

Exercise sheets

1. Sheet 1 (homework to be handed in on 04.03.25)

2. Sheet 2 (files for the exercises, homework to be handed in on 11.03.25)

3. Sheet 3 (homework to be handed in on 18.03.25)

4. Sheet 4 (homework to be handed in on 25.03.25)

5. Sheet 5 (homework to be handed in on 01.04.25)

6. Sheet 6 (homework to be handed in on 08.04.25)

7. Sheet 7 (homework to be handed in on 15.04.25)

8. Sheet 8 (homework to be handed in on 29.04.25)

9. Sheet 9 (homework to be handed in on 06.05.25)

10. Sheet 10 (files for exercises, homework to be handed in on 13.05.25)

11. Sheet 11 (homework to be handed in on 20.05.25)

12. Sheet 12 (homework to be handed in on 27.05.25)

13. Sheet 13 (homework to be handed in on 03.06.25)

14. Sheet 14

Semester Projects

In order to pass the lecture, every student needs to write a short piece of 2-5 pages on one of the following topics:

1. Efficient computation of contour trees

2. Runtime analysis of the simplex tree operations (taken)

3. The nerve theorem  (taken)

4. Čech, Delaunay, and Delaunay-Čech complexes, and their relationship.

5. Code for the Čech complex computation

6. Cellular homology (taken)

7. Persistence with integer coefficients (taken)

8. Decomposition of persistence modules and quiver representations

9. Stability with respect to the bottleneck-distance (taken)

10. The algebraic stability theorem

11. Zig-Zag peristence

12. Stability of ECC/ECP

13. Stability from point clouds to filtered complexes (taken)

14. Comparing Topotest to other available tests (taken)

15. From time series to TDA: Takens theorem and time delay/sliding window embeddings (taken)

16. Wrapping cycles in Delaunay complexes

Contact me via mail or at the lecture in order to sign up for one of them and to discuss the details.

Rules for the semester project:

After the assigningment of a project, you have 2 months to hand in the final version. The last possible date to sign up for a semester project is June 10th (the day of the last lecture).

If you have correctly presented at least one homework exercise during the exercise session, you get 1% bonus towards the semester project for each 10% of the maximal number of points  of all homework combined. (There will be 13 exercise sheets in total, 10 reachable points per exercise, 2 homework exercises per sheet, so 26 points equal 1% of the semester project.) 

Recall that the remaining rules on how to pass the class are contained in the USOS course description as well as in the slides of the first lecture.