- 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)
- 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.
- 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.
- 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)