**ERC PROJECT PHILOSOPHY OF CANONICAL QUANTUM GRAVITY**

Presentation,
Research Axes |
Practical Information |

Members | Calls for application |

Seminar "(Id)entity :: (Id)entification" |
Events |

Working Group of the Project, organisation:

Gabriel Catren, Mathieu Anel, Julien Page, Federico Zalamea (CNRS and Univ. Paris Diderot, SPHERE)

Contact: gabriel.catren((at))univ-paris-diderot.fr

WORKING GROUP ON STACKS

organised by Mathieu Anel

Programme: Fields in groupoids are generalizations of the notion of differentiable and algebraic variety. The purpose of this working group will be to understand the definition of a field including the notion of sheaf. We will follow a course for that of Bertrand Toén ( link below) .

List of things you will see :

- Varieties and beams ( Toén Chapter 1 & 2)
- Heory of homotopy groupoid ( Toén Chapter 5, Hollander, Dugger . )
- Groupoids and rustic beams down condition ( Toén end chap 5)
- Fibered categories ( Vistoli )
- Equivalence between categories and fibered in groupoids presheaves ( Hollander )
- Sheaves

December Room Gris, 734A, Tuesdays, 10:00 | ||

15/12/2015 | Mathieu Anel | Groupe de travail sur les Champs XVIII (Stacks) |

8/12/2015 | Mathieu Anel | Groupe de travail sur les Champs XVII (Stacks) |

1/12/2015 | Mathieu Anel | Groupe de travail sur les Champs XVI (Stacks) |

November Room Gris, 734A, Tuesdays, 10:00 | ||

17/11/2015 | Mathieu Anel | Working Group on Stacks XV (Stacks) |

10/11/2015 | Mathieu Anel | Working Group on Stacks XIV (Stacks) |

3/11/2015 | Mathieu Anel | Working Group on Stacks XIII (Stacks) |

October Room Gris, 734A, Tuesdays, 10:00 | ||

27/10/2015 | Mathieu Anel | Working Group on Stacks XII (Stacks) |

20/10/2015 mardi 10:00 |
Mathieu Anel | Working Group on Stacks XI (Stacks) |

13/10/2015 mardi 10:00 |
Mathieu Anel | Working Group on Stacks X (Stacks) |

June Room Alechinsky, 437A | ||

30/06/2015 Tues. 10:00 |
Mathieu Anel | Working Group on Stacks IX (Stacks) |

23/06/2015 Tues. 10:00 |
Mathieu Anel | Working Group on Stacks VIII (Stacks) |

9/06/2015 Tues. 10:00 |
Mathieu Anel | Working Group on Stacks VII (Stacks) |

2/06/2015 Tues. 10:00 |
Mathieu Anel | Working Group on Stacks VI (Stacks) |

May Room Alechinsky, 437A | ||

26/05/2015 Tues. 10:00 |
Mathieu Anel | Working Group on Stacks V |

19/05/2015 Tues. 10:00 |
Mathieu Anel | Working Group on Stacks IV |

6/05/2015 Wed. 14:30 |
Mathieu Anel | Working Group on Stacks III |

April Room Klein, 371A | ||

28/04/2015 Tues. 10:00 |
Mathieu Anel | Working Group on Stacks II |

21/04/2015 Tues. 10:00 |
Mathieu Anel | Working Group on Stacks I) |

References:

– Dugger, D., A primer on homotopy colimits

– Hollander, S., A homotopy theory for stacks

– Toën, B., A master course on algebraic stacks

– Vistoli, A., Notes on Grothendieck topologies, fibered categories and descent theory

December 2014 | |||
---|---|---|---|

12/12/2014 vendredi 14:00 |
room Mondrian, 646A | Mathieu Anel | Towards Symplectic Stacks VI |

09/12/2014 mardi 14:00 |
room Kandinsky, 631B | Mathieu Anel | Towards Symplectic Stacks V |

November 2014 (usually on Mon. and Thurs.) | |||

27/11/2014 Thurs. 10:00 |
room Mondrian, 646A | Urs Schreiber | Higher geometric quantization IVQuantization of Chern-Simons-type field theories |

24/11/2014 Mon. 14:00 |
room Gris, 734A | Mathieu Anel | Towards Symplectic Stacks IV |

!! 21 !!//11/2014 Thurs. 10:00 |
room Gris, 734A | Urs Schreiber | Higher geometric quantization IIIQuantizaton of Poisson manifolds |

17/11/2014 Mon. 14:00 |
room Gris, 734A | Mathieu Anel | Towards Symplectic Stacks III |

14/11/2014 Fri. 10:00 |
room Gris, 734A | Urs Schreiber | Higher geometric quantization IIIFormulating geometric quantization |

10/11/2014 Mon. 14:00 |
room Gris, 734A | Mathieu Anel | Towards Symplectic Stacks II |

06/11/2014 Thurs. 10:00 |
room Kandinsky, 631B | Urs Schreiber | Higher geometric quantization IBasics of higher differential geometry |

03/11/2014 Mon. 14:00 |
room Gris, 734A | Mathieu Anel | Towards Symplectic Stacks I |

03–24/11/2014

**Towards Symplectic Stacks**

by Mathieu Anel (ERC project PhiloQuantumGravity, CNRS)

The purpose of this course is to introduce the notion of "stack", which is an extension of the notion of manifold that takes care about possible symmetries of objects. Manifolds, or wannabe manifolds, are constructed (from other manifolds) by taking subspaces (defined by some equations) and/or by taking quotients (often defined by some group action). However these operations usually create singularities that prevent the result to be a manifold. We shall focus on the construction of quotients and explain how to enhance the definition of manifold into that of differentiable stack, so that it can become stable by quotients. In a second part we shall define differential forms on stacks and their symplectic structure, introducing to ideas of Toën, Pantev, Vaquié and Vezzosi. During the different lectures, we shall discuss in particular the following notions:

- Groupoids, homotopy types, classifying space of a group and cohomology
- Functor of points, moduli problems, Grothendieck topologies, sheaves and stacks
- Tangent complex, symplectic structures, symplectic groupoids

Bibliography:

- Gregory Ginot, Introduction to differentiable stacks (http://webusers.imj-prg.fr/~gregory.ginot/papers/DiffStacksIGG2013.pdf)
- Bertrand Toen, Course on Stacks (http://ens.math.univ-montp2.fr/~toen/m2.html)
- PIng Xu, Momentum maps and Morita equivalence (http://arxiv.org/pdf/math/0307319v2.pdf)
- Toen, Pantev Vaquié, Vezzosi, Shifted symplectic structure (http://arxiv.org/abs/1111.3209)

06–27/11/2014 **Higher geometric quantization**

by Urs Schreiber (Invited researcher in the context of ERC project PhiloQuantumGravity)

This lecture series begins with a basic introduction to concepts of higher (stacky) differential geometry. Then I introduce an elegant formulation of traditional geometric quantization via such concepts. As a first application, I explain a natural geometric quantization of compact Poisson manifolds, extending the familiar quantization of symplectic manifolds. I close with an outlook on aspects of the geometric quantization of Chern-Simons type field theories.

- Lecture 1: Basics of higher differential geometry
- Lecture 2: Formulating geometric quantization
- Lecture 3: Quantization of Poisson manifolds
- Lecture 4: Quantization of Chern-Simons-type field theories

Bibliography:

Notes accompanying the lectures are here.

*Philosophy of Canonical Quantum Gravity*)