Computational Topology

Lecturers: Francis Lazarus and Arnaud de Mesmay

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:

  1. Planar graphs: Jordan theorem, Euler relation, forbidden minors, Tutte embedding.

  2. Classification of surfaces and surface-embedded graphs, Euler characteristic, canonical systems of loops. Homotopy and homology, greedy generators. Homotopy test, intersection numbers.

  3. Elementary 3-dimensional topology: triangulations, knots, knot diagrams, Reidemeister moves. Normal surfaces and algorithm to test knot triviality.

  4. Simplicial complexes and limits of computational topology in high dimensions: Undecidability of the word problem, and therefore undecidability of computations in the fundamental group of a 2-complex, and undecidability of 4-manifold homeomorphism or 5-sphere recognition.

  5. Persistent homology and applications to topological data analysis.

Course notes:
  •  Introductory course
  •  Planar graphs, Exercise sheet number 1, Exercise sheet number 2
  •  Surfaces, Exercise sheet number 3, (Partial) solution
Exercise sheet number 4 is due October 19 and will be rated, Solution
Exercise sheet number 5
  •  Homotopy test, Exercise sheet number 6
  •  Minimum weight Bases Exercise sheet number 7
  •  Homology computation Exercise sheet number 8
Exercise sheet number 9 is due November 30 and will be rated
  •  Knots and 3-d computational topology
  •  Undecidability in topology
  •  Persistent homology

Suggested reading:

There are also good lecture notes available for a few other courses on Computational Topology. In particular, we recommend the ones of Éric Colin de Verdière, Jeff Erickson and Herbert Edelsbrunner. They do not necessarily cover the same topics, but might provide interesting perspectives on different facets of the subject.

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.