School Quantum Geometry
Third Erlangen Fall School on Quantum Geometry
Department Mathematik, FriedrichAlexanderUniversität ErlangenNürnberg
Erlangen, September 1518 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 indepth understanding of current research topics at the interface of mathematics and physics.
Courses
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
Abstract:
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:

 V.V. Fock, A.B. Goncharov, Dual Teichmüller and lamination spaces. arXiv:math/0510312
 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
Abstract:
In this course I intend to describe the geometric quantization procedure in all its details, starting with prequantization and going from to halfdensity quantization to halfform 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.REFERENCES:

 J.Sniatycki: Geometric quantization and quantum mechanics (Springer, 1980)
 N.Woodhouse: Geometric quantization (Oxford UP, 1980 & 1990)
 D.Simms & N.Woodhouse: Lectures on geometric quantization (LNP, Springer)
 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.

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

 Lecture notes (2nd version)
Schedule
Time  Monday Sept 15  Tuesday Sept 16  Wednesday Sept 17  Thursday Sept 18 

8:30  Registration  
9:00  Fock: Teichmüller 1 
Lechner: QFT 3 
Lechner: QFT 5 
Fock: Teichmüller 4 
10:30  Coffee break  Coffee break  Coffee break  Coffee break 
11:00  Tuynman: GeomQuant 1 
Tuynman: GeomQuant2 
Fock: Teichmüller 3 
Fock: Teichmüller 5 
12:30  Lunch break  Lunch break  Lunch break  Lunch break 
14:00  Lechner: QFT 1 
Lechner: QFT 4 
Tuynman: GeomQuant 3 
Tuynman: GeomQuant 5 
15:30  Coffee Break  Coffee Break  Coffee Break  Coffee Break 
16:00  Lechner: QFT 2 
Fock: Teichmüller 2 
Tuynman: GeomQuant 4 
Questions & Discussion 
17:00  Questions & Discussion  Questions & Discussion  Questions & Discussion  
17:30  
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, CarlThierschStr.9, 91052 Erlangen. The restaurant is in Erlangen city center and in walking distance from the hotels (Map).
Organisers
 Organisers: Catherine Meusburger, KarlHermann Neeb, Carlos Scarinci,
Department Mathematik, FAU, Cauerstrasse 11, 91058 Erlangen, Germany
 Conference secretary: Johanna Kulzer,
Department Mathematik, FAU, Cauerstrasse 11, 91058 Erlangen, Germany,
Tel: 0049 9131 85 67035, Fax: 0049 9131 85 67036,
Johanna.Kulzer@math.unierlangen.de
Travel and Directions
The workshop will take place in Erlangen, which is a university town in the southeast 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 ErlangenNürnberg, Cauerstraße 11, 91058 Erlangen. (Map, Campus Map ) The talks will take place in the lecture theatre Hörsaal 12 on the first floor.
Background
The Third Erlangen Fall School on Quantum Geometry is an activity of the Emerging Fields Project Quantum Geometry and funded by the University of ErlangenNuremberg via its Emerging Fields Initiative. Further information on the previous schools can be found here:
 First Erlangen Fall School on Quantum Geometry, October 812 2012
 Second Erlangen Fall School on Quantum Geometry, October 711 2013