**ERC PROJECT PHILOSOPHY OF CANONICAL QUANTUM GRAVITY**

To the programme of the current year.

Archives: 2015–2016, 2014–2015, 2012–2013

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

21/02/2014 - 10:00 | room Klimt, 366A | John A. Morales | Introduction toNoncommutative Geometry |

20/02/2014 - 10:00 | room Klimt, 366A | John A. Morales | Introduction toNoncommutative Geometry |

19/02/2014 - 10:00 | room Klimt, 366A | John A. Morales | Introduction toNoncommutative Geometry |

18/02/2014 - 10:00 | room Klimt, 366A | John A. Morales | Introduction toNoncommutative Geometry |

17/02/2014 - 10:00 | room Klimt, 366A | John A. Morales | Introduction toNoncommutative 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 |

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-*, Birkhaeuser, 2000.

mutative Geometry - 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-*, 793. Springer, Berlin, 1980.

matics

*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.

*Philosophy of Canonical Quantum Gravity*)