Seminar on Elementary Applied Topology


“But I don’t want to go among mad people,” Alice remarked.
“Oh, you can’t help that,” said the Cat: “We’re all mad here. I’m mad. You’re mad.”
“How do you know I’m mad?” said Alice.
“You must be,” said the Cat, “Or you wouldn’t have come here.”
Alice in Wonderland, Lewis Carroll


Math 783
Coordinator: Alessandro Valentino
Place and Time: Y27H28 | Tuesdays 15.00-17.00
Prerequisites: Linear Algebra | Geometry and Topology | Differential Geometry (optional)


If you would like to participate to the seminar, you should book the module (MAT 783) on the online platform of the University of Zurich. The first preliminary meeting will be on Tuesday, September 11th, when the various topics will be assigned.

Description

In recent years the relevance of Topology in science, engineering and computer science has rapidily increased: a few examples include applications to condensed matter theory and data analysis. The seminar provides the possibility of learning classical concepts in Topology such as complexes, homology, Morse theory, sheaves by emphasizing examples which are relevant to the current applications.

Reference

The seminar is based on the book “Elementary Applied Topology" by Robert Ghrist, which can be freely downloaded here.
Further references will be suggested according to the given topic.

Format

Each week a student will present a chapter from the book. Part of the homework is to write a short essay on the given topic, which will be then submitted at the very end of the seminar class. Attendance and partecipation to the presentations is highly suggested and encouraged.

Preliminary schedule

The following is a tentative schedule of talks, with suggested keywords.

  • Manifolds and vector fields: braids and robot planning | transversality | signals of opportunity.
    (25 Sep 2018, Stefan)
  • Simplicial and cell complexes: Vietoris-Rips complexes | Čech complexes and random samplings | decision tasks and consensus.
    (2 Oct 2018, Nicola)
  • Euler characteristic: fixed point index and population dynamics | target enumeration | Gaussian random fields.
    (9 Oct 2018, Alessandro)
  • Homology I: simplicial | cellular and singular homology | Kirchoff's current rule | reduced homology.
    (16 Oct 2018, Andreas)
  • Homology II: Čech homology | relations | functoriality and experimental imaging data.
    (23 Oct 2018, Alessandro)
  • Homotopy invariance: exact sequences | Mayer-Vietoris sequence | coverage in sensor networks | barcodes and persistent homology.
    (30 Oct 2018, Andreas)
  • Cohomology I: cochain complexes | cuts and flows | Poincaré duality and Helly's Theorem | numerical Euler integration.
    (6 Nov 2018, Erik)
  • Cohomology II: de Rham cohomology | Laplacians and Hodge Theory | circular coordinates in data sets.
    (13 Nov 2018, Nicola)
  • Morse Theory: critical points | Morse homology | Conley index | unimodal decomposition in statistics.
    (20 Nov 2018, Erik)
  • Homotopy: fundamental groups | covering spaces | braids and DNA | topological complexity of path planning.
    (27 Nov 2018, Stefan)
  • Sheaves I: cellular sheaves | logic gates | cellular sheaf cohomology | information flows.
    (04 Dec 2018, Nicola)
  • Sheaves II: topological sheaves | Euler integration | cosheaves | Bezier curves and splines.
    (11 Dec 2018, Nino)
  • Categorification: categories | clustering functors | stability in persistence | data analysis.
    (18 Dec 2018, Alessandro)