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Accueil du site > Projets de recherche financés en cours > Projet ERC Philosophie de la Gravitation Quantique Canonique > Séminaire de philosophie et physique-mathématique 2015-2016 > Seminar of Philosophy and mathematical physics 2013–2014

Seminar of Philosophy and mathematical physics 2013–2014

ERC PROJECT PHILOSOPHY OF CANONICAL QUANTUM GRAVITY

To the programme of the current year.
Archives : 2015–2016, 2014–2015, 2012–2013



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Working Group of the Project, organisation : Gabriel Catren, Julien Page, Federico Zalamea (SPHERE, CNRS, Univ. Paris Diderot)
Contact : gabriel.catren((at))univ-paris-diderot.fr


Abstracts

21/02/2014 - 10:00 room Klimt, 366A John A. Morales Introduction to Noncommutative Geometry
20/02/2014 - 10:00 room Klimt, 366A John A. Morales Introduction to Noncommutative Geometry
19/02/2014 - 10:00 room Klimt, 366A John A. Morales Introduction to Noncommutative Geometry
18/02/2014 - 10:00 room Klimt, 366A John A. Morales Introduction to Noncommutative Geometry
17/02/2014 - 10:00 room Klimt, 366A John A. Morales Introduction to Noncommutative Geometry
22/01/2014 - 10:30 room Gris, 734A Julien Page Introduction to Morita equivalence
04/12/2013 - 10:30 room Gris, 734A Dimitri Vey Gravity and topological theories, V
27/11/2013 - 10:30 room Gris, 734A Dimitri Vey Gravity and topological theories, IV
20/11/2013 - 10:30 room Gris, 734A Dimitri Vey Gravity and topological theories, III
13/11/2013 - 10:30 room Gris, 734A Dimitri Vey Gravity and topological theories, II
06/11/2013 - 10:30 salle Gris, 734A Dimitri Vey Gravity and topological theories, I
23/10/2013 - 10:30 room Gris, 734A Gabriel Catren Symplectic geometry and quantization, V
16/10/2013 - 10:30 room Gris, 734A Gabriel Catren Symplectic geometry and quantization, IV
09/10/2013 - 10:30 room Gris, 734A Gabriel Catren Symplectic geometry and quantization, III
02/10/2013 - 10:30 room Gris, 734A Gabriel Catren Symplectic geometry and quantization, II
25/09/2013 - 10:30 room Gris, 734A Gabriel Catren Symplectic geometry and quantization, I


Abstracts :


22/01/2014 Introduction to noncommutative geometry
by John Alexander Cruz Morales- (Instituto Nacional de Matematica Pura e Aplicada, IMPA, Rio de Janeiro)
In this course I will give a brief introduction to the noncommutative geom- etry in Connes’ sense. In the -rst 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, Grupoids and Morita equivalence. The goal in this part is try to discuss the importance of grupoids in the framework of the 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.


Bibliography :

  • A. Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  • J. M. Gracia-Bondia, J. C. Varilly, and H. Figueroa, Elements of Noncom- mutative Geometry, Birkhaeuser, 2000.
  • N. Higson, and J. Roe, Analytic K-homology, Oxford Mathematical Mono- graphs. 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. Renaul, "A groupoid approach to C*—algebras", Lecture Notes in Mathe- matics, 793. Springer, Berlin, 1980.





This project has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) for research, technological development and demonstration under grant agreement n° 263523 (Project Philosophy of Canonical Quantum Gravity)





06/11/2013–04/12/2013 Introduction to Morita equivalence
by Julien Page
We will propose an introduction to Morita equivalence. This notions appears mainly in the following subjects : 1) Ring (and associative algebras) representation theory ; 2) C*-algebras representation theory ; 3) the theory of symplectic realizations of Poisson manifolds. Thus, two rings A and B are said to be Morita equivalent if their associated categories of left modules are equivalent. We will show how this notion, invented for rings during the ’50s by the japanese mathematician Morita, extends naturally to k-algebras, C*-algebras and Poisson manifolds. Finally, we will present some of the conceptual and technical problems at stake here.
We will suppose familiarity with the notions of ring, algebra, A-module, C*-algebra, Poisson manifold and equivalence of categories.


Bibliography :

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


06/11/2013–04/12/2013 Gravity and topological theories
by Dimitri Vey
In recent decades, a plethora of actions and theories (Einstein- Hilbert Palatini, Einstein- Cartan , Plebanski , MacDowell -Mansouri , BF theories, Chern -Simons...) have emerged as possible formulations and variations of the gravitational theory. Entanglement with topological terms (Holst, Euler, Pontryagin, Nieh-Yan), the notions of Hodge duality and self-duality are cornerstones of these constructions. The objective of this course is to make a taxonomy of the different useful actions for the theory of gravity as well as their associated mathematical objects (canonical variables, Hodge duality,etc.). We will focus on :
I- The relationship between the different formulations.
II- The underlying guiding idea in the construction of the gravitational action principles.
The goal is to obtain a hierarchy of these different formulations. In particular, we will explore the tension between topological theories and the additional specific constraints on them that allow us to recover a gravitational theory (with gravitational local degrees of freedom).


Bibliography

  • Baez, J.C., "An introduction to spin foam models of BF theory and quantum gravity, in Geometry and Quantum Physics (Schladming, 1999)", Lecture Notes in Phys., Vol. 543, Editors H. Gausterer, H. Grosse, Springer, Berlin, 2000, 25-93, gr- qc/9905087
  • Cattaneo, A.S., et al., "Topological BF theories in 3 and 4 dimensions", J. Math. Phys. 36 (1995), 6137-6160, hep-th/9505027.
  • Freidel, L. and Starodubtsev, A., Quantum gravity in terms of topological observables
  • Krasnov, K., Plebanski Formulation of General Relativity : A Practical Introduction.
  • Wise, D.K., "MacDowell-Mansouri gravity and Cartan geometry", Classical Quantum Gravity 27 (2010), 155010, 26 pages, gr-qc/0611154.
  • Zanelli J., Chern-Simons Forms in Gravitation Theories


25/09/2013–23/10/2013 Symplectic geometry and quantization
by Gabriel Catren
The objective of this course is to analyze the relations between three fundamental concepts of mechanics, namely the concepts of state, observable, and operator. To do so, we shall introduce the basics of symplectic geometry paying special attention to the moment map introduced by J.-M. Souriau, the Mardsen-Weinstein’s symplectic reduction formalism and its cohomological reformulation (namely, the BRST cohomology). We shall then analyze the role played by these classical geometric structures in the quantization of the corresponding classical systems. In order to do this, we shall introduce Kirillov’s orbit method (that describes the unitary representations of nilpotent Lie groups) and – if we have enough time – the geometric quantization formalism (developed by B. Kostant, J.-M. Souriau, and A.A. Kirillov).


Bibliography
– Abraham, R. & Marsden, J.E. [1978] : Foundations of mechanics, Massachusetts : Addison-Wesley Publishing Company, Inc., 1978.
– Brylinski, J.L. [1993] : Loop spaces, characteristic classes, and geometric quantization, Boston : Birkhäuser Boston, Program of Mathematics, 107.
– Figueroa-O’Farrill, J.M. [1989] : BRST Cohomology and its Applications to Two Dimensional Conformal Field Theory, PhD dissertation.
– Guillemin, V. and Sternberg, S. [1982] : Geometric Quantization and Multiplicities of Group Representations, Invent. math., 67, 515-538.
– Henneaux, M. and Teitelboim, C. [1994] : Quantization of gauge systems, Princeton Univ. Press.
– Kirillov, A.A. [2004] : Lectures on the Orbit Method, Graduate Studies in Mathematics, Vol. 64, AMS.
– Kostant, B. [1970] : Quantization and Unitary representations, Lecture Notes in Mathematics, Vol. 170, Berlin-Heidelberg-New York : Springer-Verlag.
– Kostant, B. and Sternberg, S. [1987] : Symplectic Reduction, BRS Cohomology, and Infinite Dimensional Clifford Algebras, Annals of Physics 176, 49-113.
– Landsman, N.P. [1998] : Mathematical Topics Between Classical and Quantum Mechanics, Springer Monographs in Mathematics, Springer-Verlag, New York.
– Marsden, J.E. and Ratiu, T.S. [1999] : Introduction to Mechanics and Symmetry, second ed., Springer-Verlag, New York.
– Souriau, J.M. [1997] : Structure of dynamical systems. A symplectic view of physics. Boston : Birkhäuser, 1997.
– Woodhouse, N. [1992] : Geometric quantization, Oxford : Oxford University Press.
















This project has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) for research, technological development and demonstration under grant agreement n° 263523 (Project Philosophy of Canonical Quantum Gravity)