Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

Introduction to Topological Data Analysis (2024)

Lecturers: Patrick Schnider (OAT Z 13.1) Simon Weber (OAT Z 18), contact assistant Shengzhe Wang Thu 13-14, CAB G 51, Fri 12-14, CAB G 61 Wed 10-12, CHN F 46 English Fundamental concepts, techniques and results in Topological Data Analysis. Preliminary list of topics: Mathematical Background (Topology, Simplicial complexes, Homology) Persistent Homology Complexes on point clouds The TDA pipeline Reeb Graphs Mapper Algorithm The complete lecture notes from last year can be found here: [L] Lecture Notes. These are a collection of the scribe notes from each lecture of 2023. The up-to-date chapters for this year's edition are published here as the course progresses: We additionally suggest the following supporting textbooks: [CTDA] Computational Topology for Data Analysis by Tamal K. Dey, Yusu Wang [Comp.] Computational Topology: An Introduction by Herbert Edelsbrunner, John Harer [App.] Topological Data Analysis with Applications, by Gunnar Carlsson, Mikael Vejdemo-Johansson [EAT] Elementary Applied Topology by Robert Ghrist [AT] Algebraic Topology by Allen Hatcher The course assumes knowledge of discrete mathematics, algorithms and data structures and linear algebra, as supplied in the first semesters of Bachelor Studies at ETH.
 Special Assignment 1: Special Assignment 2: 21.3. - 11.4. 12:15 Exercise Sheet 1 2.5. - 17.5. 12:15 Exercise Sheet 2

Absence. If there are compelling reasons why you cannot hand in a special assignment on the due date foreseen, please contact us post-haste and we can look for a solution. Note, however, that we will grant requests of this type only in very exceptional cases (doctor's note, military service, funeral, etc.). Private events (vacation, sports, career fairs, etc.) are never sufficient grounds. As far as special assignments are concerned, you can hand them in to the contact assistant by email at any time before the due date, so there is no need to be present in person. Regarding the final exam, ETH regulations apply.

Exchange students. Exchange students enrolled for TDA in spring might want to return to their home university prior to the date of the final exam. If this applies to you and you nonetheless wish to be evaluated and given credits for TDA, then please take a look at the following website for current regulations.

Schedule

In the table below you can find the lecture dates and the preliminary topics. The exercises and their solutions will be published here.

 Date Content Exercises and links Relevant chapters #1 22.02.2024 Information about the course, Intro to Topological Spaces Slides, Colab [L] 1 #2 23.02.2024 Mathematical Background Exercises 1 (Video for Exercise 6) [L] 2.1, 2.2, 2.3 #3 29.02.2024 Algebra [L] 2.4 #4 01.03.2024 Simplicial Complexes, Intro to Homology Slides, Exercises 2 [L] 3.1, 3.2.1, 3.2.2 #5 07.03.2024 Homology [L] 3.2.1-3.2.3 #6 08.03.2024 Homology Exercises 3 Colab [L] 3.2.4-3.2.6 #7 14.03.2024 Homology of spheres [L] 3.2.7 #8 15.03.2024 Induced Homology, Brouwer fixed-point theorem Exercises 4 [L] 3.2.8, 3.2.9 #9 21.03.2024 Persistent Homology Slides [L] 4.1, 4.2 #10 22.03.2024 Persistent Homology, Algorithms SlidesExercises 5 [L] 4.3 #11 28.03.2024 Complexes on Point Clouds Slides [L] 5.1, 5.2 #12 11.04.2024 Complexes on Point Clouds (cont'd) Exercises 6 SPA 1 [L] 5.2, 5.3 #13 12.04.2024 Bottleneck distance, Stability [L] 6.1.1 #14 18.04.2024 Wasserstein distance Exercises 7 Colab [L] 6.1.2 #15 19.04.2024 Guest Lecture: Bastian Rieck (Helmholtz München) Slides #16 25.04.2024 Guest Lecture: Bernadette Stolz (EPFL) Exercises 8 #17 26.04.2024 Interleaving Distance [L] 6.2 #18 02.05.2024 Stability for Cech Complexes Exercises 9 Colab [L] 6.2.3 #19 03.05.2024 Guest Lecture: Marius Huber (UZH) #20 10.05.2024 Interval Decompositions, Reeb graphs Exercises 10 Slides [L] 6.3, 7.1, 7.3 #21 16.05.2024 Homology of Reeb graphs [L] 7.1.1 #22 17.05.2024 Interleaving Distance for Reeb graphs Exercises 11 Colab SPA 2 [L] 7.2.1 #23 23.05.2024 Multi-scale Mapper [L] 7.4 #24 24.05.2024 Optimal Generators Exercises 12 Slides [L] 8.1, 8.2 #25 30.05.2024 #26 31.05.2024

Procedures, Exercises, Exam

Every week we provide you with exercises. The students are split into small groups, and the members of each group work together. At the end of the session, for each exercise, a student from a group presents their solution to the rest of the students. In addition to the exercise sessions, we encourage you to solve the exercises in written form and to hand in your solutions to the teaching assistant. Your solutions are thoroughly commented, but they do not count towards your final grade. The motivation to work on the exercises stems from your interest in the topic (and possibly also the desire to succeed in the exam).