Master's course (2014/15, spring semester) - Topological Quantum Field Theories

This course is an advanced course of the Master Program (spring semester 2014-15) at the University Lille 1 (France)

by Alexis Virelizier


Overview | Syllabus | Exercices | Requirements | Bibliography | Useful information and links


Overview

Topological quantum field theories (TQFTs) originate ideas of quantum physics. Since then, the concept has grown considerably in many directions, and had an important influence in mathematics, including topology in small dimension. In particular, TQFTs form a framework that organizes some topological invariant of manifolds in a structure which can not be captured by the standard techniques of algebraic topology.

The aim of the course is to introduce the notion of TQFT from the mathematical viewpoint and to consider some examples, especially in dimension 2 and 3. We will also discuss some recent developments concerning extended TQFTs, which are closely related to higher categories.

This course builds on the fundamental Master's course Homology and Topology given on the previous semester (fall semester 2014/2015) by Ivo Dell'Ambrogio and Benoit Fresse, and opens on various actual research topics (fusion categories, extended TQFTs, etc ...).


Syllabus

§1. Introduction to TQFTs

§2. TQFTs of dimension 2

§3. TQFTs of dimension 3

§4. Extended TQFTs


Exercises


Requirements

It is recommended to have followed the course Homology and Topology given the previous semester, or otherwise an equivalent course.


Bibliography

  1. Atiyah, M., Topological quantum field theories, Publications Mathématiques de l'IHÉS 68 (1989), 175-186.
  2. Etingof, P., Nikshych, D., Ostrik, V., On fusion categories, Annals of Mathematics 162 (2005), 581-642.
  3. Koch, J., Frobenius algebras and 2D topological quantum field theories, London Mathematical Society Student Texts, 59. Cambridge University Press, Cambridge, 2004.
  4. MacLane, S., Categories for the working mathematician, Second edition, Springer-Verlag, New York, 1998.
  5. Schommer-Pries, C., The Classification of Two-Dimensional Extended Topological Field Theories, Thèse de Doctorat, 2009.
  6. Turaev, V., Quantum invariants of knots and 3-manifolds, Second revised edition, de Gruyter Studies in Mathematics, 18. Walter de Gruyter, Berlin, 2010.

Useful information

Dates and times:
The course will take place from January to April 2015 and is granted 12ECTS credits.

Location and transport:
The course will be held at the Faculty of Mathematics of the University Lille 1, campus "Cité Scientifique", Lille - Villeneuve d'Ascq, see here.
The campus "Cité Scientifique" is connected to the main train stations of Lille and to the center of the city by an efficient metro line (count 20 minutes from the train stations to get to the math building). To get the train station of Lille, it takes 1h from Paris (by TGV), 35min from Brussels (by TGV), and 1h20 from London (by Eurostar).

Links: