Introduction to quantum topology

Lecture at the Master Class in Geometry, Topology and Physics (2016-17) at the University of Geneva (Switzerland)

by Alexis Virelizier

Description

The aim of the course is to introduce the notions of quantum invariants and TQFTs from the mathematical viewpoint and to consider some examples, especially in dimension 2 and 3.

Topics

1. TQFTs in dimension 2
2. The Jones polynomial
3. Quantum groups and the Drinfeld double
4. The Turaev-Viro quantum invariants of 3-manifolds

Prerequisites

Basic notions of algebra (algebras and their modules, tensor product) and of algebraic topology (fundamental group, homology).

Exercise sheets

Exercise sheet 1, pdf.
Exercise sheet 2, pdf.
Exercise sheet 3, pdf.
Exercise sheet 4, pdf.
Exercise sheet 5, pdf.
Exercise sheet 6, pdf.
Exercise sheet 7, pdf.
Exercise sheet 8, pdf.
Exercise sheet 9, pdf.
Exercise sheet 10, pdf.

Videos

The videos of the lectures can be found here.

Bibliography

[1] Kock, J., Frobenius algebras and 2D topological quantum field theories. London Mathematical Society Student Texts, 59. Cambridge University Press, Cambridge, 2004.
[2] Lickorish, W. B. R., An introduction to knot theory. Graduate Texts in Mathematics, 175. Springer-Verlag, New York, 1997.
[3] Kassel, C., Quantum groups. Graduate Texts in Mathematics, 155. Springer-Verlag, New York, 1995.
[4] Turaev, V., Quantum invariants of knots and 3-manifolds. Second revised edition. de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co., Berlin, 2010.
[5] Ohtsuki, T., Quantum invariants. A study of knots, 3-manifolds, and their sets. Series on Knots and Everything, 29. World Scientific Publishing Co., Inc., River Edge, NJ, 2002.