## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Introduction to Topological Data Analysis (2023)

Lecturers: Patrick Schnider (OAT Z 13.1) Simon Weber (OAT Z 18), contact assistant Anton Künzi 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 lecture notes can now be found here: Lecture Notes These are a collection of the scribe notes from each lecture, see below. We will also list the relevant chapters in the textbooks below (using their abbreviations). The main textbook for the course is: [CTDA] Computational Topology for Data Analysis by Tamal K. Dey, Yusu Wang Supporting textbooks: [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: 23.03. - 06.04. (Deadline is at 13:15) Exercise Sheet 1 04.05. - 19.05. (Deadline is at 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 Scribe notes Relevant book chapters #1 23.02.2023 Information about the course, Intro to Topological Spaces Slides Scribe 1 [CTDA] 1.1 #2 24.02.2023 Mathematical Background Exercises 1 Scribe 2 [CTDA] 1.1, 1.2, 1.3 #3 02.03.2023 Some Algebra Scribe 3 [CTDA] 2.4 #4 03.03.2023 Simplicial Complexes Slides, Exercises 2 Scribe 4 [CTDA] 2.1, 2.2 #5 09.03.2023 Chains and Boundaries Scribe 5 [CTDA] 2.4, 2.5 #6 10.03.2023 Homology Exercises 3 Scribe 6 [CTDA] 2.5 #7 16.03.2023 Homology of spheres Scribe 7(updated on 17.03.!) #8 17.03.2023 Induced Homology, Brouwer fixed-point theorem, Intro to Persistence Exercises 4 Scribe 8(updated on 22.03.!) [CTDA] 2.5.1 #9 23.03.2023 Persistent Homology Slides Scribe 9 [CTDA] 3.1, 3.2 #10 24.03.2023 Persistent Homology, Algorithms SlidesExercises 5 Scribe 10 [CTDA] 3.1, 3.2 #11 30.03.2023 Complexes on Point Clouds (online) Slides [CTDA] 2.2, 2.3 #12 31.03.2023 Tutorial on Programming in TDA Colab Exercises 6 #13 06.04.2023 Guest Lecture by Bastian Rieck on TDA in Machine Learning Slides Recording #14 20.04.2023 Complexes on Point Clouds (cont'd) Exercises 7 Scribe 11&14 [CTDA] 2.2, 2.3 #15 21.04.2023 Distance Measures and Stability Exercises 8 Scribe 15 [CTDA] 3.2.1, [Comp.] VIII.2 #16 27.04.2023 Wasserstein Distance Scribe 16 [CTDA] 3.2.1 #17 28.04.2023 Interleaving Distance Exercises 9 Scribe 17 [CTDA] 3.4 #18 04.05.2023 Interval Modules, Structure Theorem Exercises 10 Scribe 18 [CTDA] 3.4 #19 05.05.2023 Reeb Graphs Slides Scribe 19 [CTDA] 7 #20 11.05.2023 Distance Measures for Reeb Graphs Exercises 11 Scribe 20 #21 12.05.2023 Functional distortion distance, Mapper Colab (Mapper demo) Scribe 21 #22 19.05.2023 Guest Lecture: Marius Huber (UZH) Slides, Jupyter Notebook #23 25.05.2023 Multi-scale Mapper TDA Challenge Scribe 23 [CTDA] 9.3 #24 26.05.2023 Optimal Generators Exercises 12 Slides Scribe 24 [CTDA] 9.2 #25 01.06.2023 Applications Slides Scribe 25 #26 02.06.2023 Applications Scribe 26

## 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).