Seminar

  • Towards Symplectic Stacks
    by Mathieu Anel

Abstract: 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:

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

Towards Symplectic Stacks
Session Date Room
Towards Symplectic Stacks VI  12/12/2014 – 14:00 Mondrian, 646A
Towards Symplectic Stacks V  9/12/2014 – 14:00 Kandinsky, 631B
Towards Symplectic Stacks IV  24/11/2014 – 14:00 Gris, 734A
Towards Symplectic Stacks III  17/11/2014 – 14:00 Gris, 734A
Towards Symplectic Stacks II  10/11/2014 – 14:00 Gris, 734A
Towards Symplectic Stacks I  03/11/2014 – 14:00 Gris, 734A

Bibliography:

Ginot G., Introduction to differentiable stacks (2013)
Toen, B., Course on Stacks (2006-2007)
Xu, P., Momentum maps and Morita equivalence (2004)
Pantev, T., et al, Shifted Symplectic Structures (2013)

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  • Higher geometric quantization
    by Urs Schreiber

Abstract: 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: Quantizaton of Poisson manifolds
Lecture 4: Quantization of Chern-Simons-type field theories

Higher geometric quantization
Session Date Room
Higher geometric quantization IV  27/11/2014 – 10:00 Mondrian, 646A
Higher geometric quantization III  21/11/2014 – 10:00 Kandinsky, 734A
Higher geometric quantization II  14/11/2014 – 10:00 Gris, 734A
Higher geometric quantization I  06/11/2014 – 10:00 Kandinsky, 631B

Bibliography:
Notes accompanying the lectures can be found here.

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  • Introduction to Noncommutative Geometry, Feb. 17-21, 2014
    by John Alexander Cruz Morales (Instituto Nacional de Matematica Pura e Aplicada, IMPA, Rio de Janeiro)

Abstract: In this course I will give a brief introduction to noncommutative geometry in Connes’ sense. In the first part, I will start presenting a correspondence between algebra and geometry in the context of Gelfand-Naimark theorem and then introduce the notion of noncommutative quotients, Groupoids and Morita equivalence. The goal in this part is to try to discuss the importance of groupoids in the framework of noncommutative geometry. In the second part, I will present a general discussion about Atiyah-Singer index theorem and some ideas of K-theory. This will serve as a preparation for the presentation of the cyclic (co)homology and Chern-Connes character in the third part of the course.
The course will be as self-contained as possible. Questions, comments and discussions during the sessions will be very welcome and encouraged.

Introduction to Noncommutative Geometry 
Session Date room
Introduction to Noncommutative Geometry V 21/02/2014 – 10:00 Klimt, 366A
Introduction to Noncommutative Geometry IV 20/02/2014 – 10:00 Klimt, 366A
Introduction to Noncommutative Geometry III 19/02/2014 – 10:00 Klimt, 366A
Introduction to Noncommutative Geometry II 18/02/2014 – 10:00 Klimt, 366A
Introduction to Noncommutative Geometry I 17/02/2014 – 10:00 Klimt, 366A

Bibliography:

A. Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
J. M. Gracia-Bondia, J. C. Varilly, and H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser, 2000.
N. Higson, and J. Roe, Analytic K-homology, Oxford Mathematical Monographs. Oxford Science Publications. Oxford University Press, Oxford, 2000.
M. Karoubi, Homologie cyclique et K-thorie, Astrisque No. 149, 1987.
Masoud Khalkhali, Very Basic Noncommutative Geometry
J. Renault, A groupoid approach to C*—algebras, Lecture Notes in Mathematics, 793. Springer, Berlin, 1980.

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  • Introduction à l’équivalence de Morita
    by Julien Page

Abstract: On proposera une introduction à l’équivalence de Morita. Celle-ci apparaît notamment dans : 1) la théorie des représentations des Anneaux (et des Algèbres associatives) ; 2) la théorie des représentations des C*-Algèbres ; 3) la théorie des réalisations symplectiques des variétés de Poisson. Ainsi, deux anneaux A et B sont dit Morita équivalents si leur catégorie de modules à gauche A-Mod et B-Mod sont équivalentes. On montrera comment cette notion inventée dans les années 1950 par le mathématicien japonais Morita pour les anneaux, s’est naturellement étendue aux k-algèbres, aux C*-algèbres et aux variétés de Poisson. Et on présentera quelques enjeux conceptuels et problèmes techniques d’une telle équivalence.
Supposés connus : Les définitions d’un anneau, d’une algèbre, d’un A-module, d’une C*-algèbre, d’une variété de Poisson, d’une équivalence de catégories.

Introduction à l’équivalence de Morita 
Session Date Room
Introduction à l’équivalence de Morita I 22/01/2014 – 10:30 Gris, 734A

Bibliography:

Masoud Khalkhali, Basic Noncommutative Geometry, European Mathematical Society (2009)
Ping Xu, Morita Equivalence of Poisson Manifolds, Comm. Math. Phys. 142, 493-509 (1991)

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