Where and when: See timetables.
Prerequisites: Nothing strictly required. Basics in graph theory and algorithms will help, as well as a certain taste for mathematics.
Objective: Computational topology is primarily concerned with the development of efficient algorithms for solving topological problems. This course is an introduction to the main tools and concepts in the field. While topology is an old and mature mathematical field, the study of its effective aspects has only started to flourish in the last decades. We will start with graph theory using planar and surface-embedded graphs to introduce fundamental topological notions as we progress. We then increase the dimension progressively and finish with persistence theory, a blooming topological tool in the analysis of big data.
Tentative plan:
Course notes: ,
• Introductory course | ||
• Planar graphs, | Exercise sheet 1 | |
Exercise sheet 2 | ||
• Surfaces, | Exercise sheet 3, (Partial) solution | |
Exercise sheet 4 is due October 19 and will be rated, Solution | ||
Exercise sheet 5 | ||
• Homotopy test, | Exercise sheet 6 | |
• Minimum weight Bases | Exercise sheet 7 | |
• Homology computation | Exercise sheet 8 | |
Exercise sheet 9 is due November 30 and will be rated | ||
• Knots and 3-d computational topology | Exercise sheet 10 | |
Exercise sheet 11 | ||
Exercise sheet 12 is due January 12 and will be rated | ||
• Undecidability in topology | ||
• Persistent homology |
Validation: Homework, and work (written report and oral presentation) on a research article. The rules are: three homeworks will be rated and we will consider the average HS of the two best scores. The final score will be obtained by the formula (HS+OS)/2, where OS is the score for the oral presentation.
Oral presentation: You have to select one project in the list below. We recommend that you choose a partner and work by pairs on a project. Each project should be chosen by at most one pair. Try to avoid conflict by discussing your choice with the whole class. The presentation should last 25 mn followed by questions and a short discussion. You may use your own laptop or bring your presentation on a usb key. In this latter case, we ask you to use a pdf file. Each member of a pair should speak approximately the same amount of time. The rough idea is to present the main ideas in the chosen article(s), give some details and say one proof that you find worth be presented. We do not require a written report for the project, but we will ask for an electronic copy of your slides. Some papers are only accessible via your institution. We nonetheless link each paper to a public version so that you get an idea of the content. When indicated, you should download the final version in the end. Please, send us an email if you have any problem to download a paper.