School Quantum Geometry

Third Erlangen Fall School on Quantum Geometry

Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg

Erlangen, September 15-18 2014

The Erlangen Fall Schools on Quantum Geometry are a series of interdisciplinary schools on mathematical physics topics surrounding geometry and quantisation. They address postdocs and PhD students from both subjects and aim to provide an overview as well as an in-depth understanding of current research topics at the interface of mathematics and physics.


The Third Erlangen Fall School on Quantum Geometry involves three intensive lecture series. Each lecture series consists of approximately 6 hours of lectures as well as problem and discussion sessions. They are aimed at a mixed audience of mathematicians and physicists and are accessible to PhD students and postdocs.

  • Teichmüller Theory

    Vladimir Fock, Université de Strasbourg, IRMA

    The main aim of the course is to present a combinatorial approach to Teichmüller spaces of complex structures on Riemann surfaces. We will give an explicit coordinate description of this space using hyperbolic geometry and then proceed to its quantization using quantum dilogarithm. We will also discuss relations of Teichmüller spaces to cluster varieties, Dehn invariant and Bloch group, higher Teichmüller spaces. If time permits the relation to integrable systems will also be discussed. Knowledge of these subjects is not assumed. On the contrary, we hope that the course will serve as an introduction to these branches of mathematics.

    A significant part of the course can be found in the following articles:

      1. V.V. Fock, A.B. Goncharov, Dual Teichmüller and lamination spaces. arXiv:math/0510312
      2. L.Chekhov, V.V.Fock, Quantum Teichmüller space, arXiv:math/9908165


    Elementary knowledge of hyperbolic geometry in two and three dimension and of the notion of Poisson structure is recommended. We also recommend to look through the definitions of Weil representation, Dehn invariant and (Liouville) integrable system.


  • Deformations of Operator Algebras and the Construction of Quantum Field Theories

    Gandalf Lechner, Universität Leipzig


  • Geometric Quantization

    Gijs Tuynman, Université Lille 1

    In this course I intend to describe the geometric quantization procedure in all its details, starting with prequantization and going from to half-density quantization to half-form quantization. I will try to emphasize the why of all constructions, starting with the initial motivation from physics for this procedure. I will also give explicit formulas that permit to compute examples. Depending upon on time I will sketch most/some of the proofs.


      1. J.Sniatycki: Geometric quantization and quantum mechanics (Springer, 1980)
      2. N.Woodhouse: Geometric quantization (Oxford UP, 1980 & 1990)
      3. D.Simms & N.Woodhouse: Lectures on geometric quantization (LNP, Springer)
      4. J.-M.Souriau: Structure of dynamical systems (Birkhäuser) = Structure des systèmes dynamiques (Dunod, 1970)


    In my course I will rely upon basic knowledge concerning symplectic mechanics, Lie groups and fibre bundles (but just the basics). The following references contain certainly (infinitely) much more than I will need. In any case, I will recall (if necessary) what I need.

      1. R.Abraham & J.Marsden: Foundations of mechanics
      2. J.-M.Souriau: Structure of dynamical systems (see above)
      3. P.Libermann & C.-M.Marle: Symplectic geometry and analytical mechanics (Kluwer)
      4. Warner: Foundations of differentiable manifolds and Lie groups (Springer GTM)
      5. Husemoller: Fibre bundles (Springer GTM)


  • Lecture notes (2nd version)


Time Monday Sept 15 Tuesday Sept 16 Wednesday Sept 17 Thursday Sept 18
8:30 Registration
9:00 Fock:
Teichmüller 1
Teichmüller 4
10:30 Coffee break Coffee break Coffee break Coffee break
11:00 Tuynman:
GeomQuant 1
Teichmüller 3
Teichmüller 5
12:30 Lunch break Lunch break Lunch break Lunch break
14:00 Lechner:
GeomQuant 3
GeomQuant 5
15:30 Coffee Break Coffee Break Coffee Break Coffee Break
16:00 Lechner:
Teichmüller 2
GeomQuant 4
Questions & Discussion
17:00 Questions & Discussion Questions & Discussion Questions & Discussion
19:00 Conference Dinner
@ Unicum

Lunch Breaks

Lunch will be provided at the university restaurant in the same building. There will be a choice of different dishes, including meat or fish, vegetarian options and a salad bar.

Conference Dinner

The conference dinner will take place Wednesday, September 17 at 7:00 pm at

the restaurant Unicum, Carl-Thiersch-Str.9, 91052 Erlangen. The restaurant is in Erlangen city center and in walking distance from the hotels (Map).


Travel and Directions

The workshop will take place in Erlangen, which is a university town in the south-east of Germany in the region of Franconia. It is easy to reach by a combination of air and train travel. The conference venue is the Department of Mathematics, University of Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen. (Map, Campus Map ) The talks will take place in the lecture theatre Hörsaal 12 on the first floor.


The Third Erlangen Fall School on Quantum Geometry is an activity of the Emerging Fields Project Quantum Geometry and funded by the University of Erlangen-Nuremberg via its Emerging Fields Initiative. Further information on the previous schools can be found here:

Friedrich-Alexander-Universität Erlangen-Nürnberg