GAP4: Aspects of Quantization Preliminary Program ------------------------- Monday, 12/June/2006 9:00 Registration + Opening 9:40 V. Ginzburg I, Geometric quantization, localization and cobordisms of Hamiltonian torus actions 10:50 V. Ginzburg II Lunch 14:00 B. Tsygan I 15:10 B. Tsygan II 16:20 Talks by people from Hanoi University of Education 17:20 Talks by people from Hanoi University of Education Tuesday, 13/June/2006 8:30 R. Sjamaar I, Equivariant index theory and quantization 9:40 V.N. San I, Quantum Birkhoff normal forms and spectral asymptotics 10:50 V.N. San II Lunch 14:00 M. Audin I, Lagrangian submanifolds 15:10 M. Audin II 16:20 R.L. Fernandes, Symplectization commutes with reduction 17:20 D.N. Diep, Geometric quantization and construction of irreducible unitary representations Wednesday, 14/June/2006 8:30 V. Ginzburg III 9:40 B. Tsygan III 10:50 S. Merkulov I, PROPs, graph complexes and deformation quantization Lunch 14:00 R. Sjamaar II 15:10 R. Sjamaar III Excursion in Hanoi Thursday, 15/June/2006 8:30 M. Audin III 9:40 S. Merkulov II 10:50 S. Merkulov III Lunch 14:00 V.N. San III 15:10 T. Holm, Orbifold cohomology of Abelian symplectic reduction 16:10 N. Ciccoli, Duality in Hochschild homology for quantum groups 17:10 N.V. Hai, Deformation quantization and representations of some Lie groups 17:40 T.D. Dong, Mp^c structures and geometric quantization on U(1)-covering. Friday, 16/June/2006 8:30 A. Wade, Remarks on deformation of generalizaed complex structures 9:30 D.V. Duc, Tetrads as key development tool of Einstein-Cartan-Evans (ECE) theory 10:30 H. Bursztyn, Moent maps and pure spinors 11:30 B. Davis, Dirac fiber bundles Lunch 14:20 G. Ginot, Loop product for orbifolds 15:20 I. Waschkies, Simple sheaves on smooth complex Lagrangian submanifolds 16:20 C. Laurent, Differential gerbs and connections 17:20 M. Zambon, Coisotropic embeddings in Poisson manifolds Saturday 17/June/2006 Excursion to Halong Bay. Night at Halong Bay Sunday 18/June/2006 Return from Halong Bay in the afternoon. Back in Hanoi at about 18:00. ........ Abstracts: Minicourses: Michèle Audin: Lagrangian submanifolds Abstract: TBA -- Viktor Ginzburg: Geometric quantization, localization and cobordisms of Hamiltonian torus actions Abstract: We start these lectures with a discussion of geometric quantization and address the question why the geometric quantization of a manifold is a virtual object rather than a genuine vector space or a representation. We show that, as a consequence of a localization theorem, the geometric quantization of a Hamiltonian torus action can be expressed as the sum of geometric quantizations of the tangent spaces at the fixed points. Then, turning to the main subject of the lectures, we introduce the structures necessary to state the linearization theorem asserting that the underlying manifold is itself equivalent in a certain sense to the sum of these tangent spaces. The equivalence relation required here is a non-trivial notion of cobordism of Hamiltonian torus actions on manifolds which are not necessarily compact. These talks are based on the speaker's joint work with Victor Guillemin and Yael Karshon. -- Sergei Merkulov: PROPs, graph complexes and deformation quantization Abstract: We shall give an introduction to the theory of operads and PROPs and discuss in detail a rather surprising link between graph complexes and differential-geometric structues. A particular attention will be paid to the PROP profiles of Poisson and Nijenhuis structures, as well as to the operadic graph complex behind torsion-free affine connections. As an application of this new approach to geometric structures we give a short proof of Konstevich´s theorem on quatization of arbitrary Poisson structures in R^n. --- Vu Ngoc San: Quantum Birkhoff normal form and spectral asymptotic The Birkhoff normal form, in classical mechanics, is a well known refinement of the averaging method. It consists in performing a suitable canonical (symplectic) transformation in order to simplify the Hamiltonian at a formal level. I will present a quantum version of this normal form which contains the classical one as a limit when Planck's constant $\hbar$ tends to zero. Using the calculus of pseudo-differential operators, this normal form gives a very precise approximation of the quantum spectrum. In case of a classical periodic flow one can perform a quantum analogue of the symplectic reduction of Weinstein and Marsden, yielding even more precise spectral asymptotics. From a molecular viewpoint, this gives a description of the fine structure of polyads close to the bottom of the spectrum. My talk will be based on a joint work with Laurent Charles. ---- Reyer Sjamaar: Equivariant index theory and quantization The index theory of prequantizable symplectic manifolds has many of the desirable formal features of a quantization procedure. For example, in the presence of a group action the index of a symplectic manifold M is a (virtual) representation of the group, and a theorem due to Meinrenken and others asserts that the isotypical subspaces of this representation are the indices of the symplectic quotients of M ("quantization commutes with reduction"). In my three lectures I will explain this point of view, discuss the requisite symplectic geometry and go into some more recent developments. -- Boris Tsygan: TBA -- Talks: Henrique Bursztyn: Moment maps and pure spinors -- Nicola Ciccoli: Duality in Hochschild homology for quantum groups (joint work with U. Khramer) The idea is to explain the duality between Hochschild homology and cohomology with coefficients, as recently described by v.d.Bergh and Brown-Zhang, on quantum groups in terms of their semiclassical counterpart. After introducing a spectral sequence like in Feng-Tsygan 1990 paper one can show the role of the modular class in this duality. An outcome is an explanation of quantum dimension drop, and of the role of twisted Hochschild homology for quantum groups, purely in terms of Poisson geometry. -- Ben Davis: Dirac fiber bundles. Given a Dirac fiber bundle with connection over a Dirac base we extend the Dirac structures of base and fiber to the total space. Furthemore, we investigate the case of principle G-bundles and ask when the construction of Dirac structures is natural with respect to the formation of associated bundles." -- Do Ngoc Diep: Geometric Quantization and Constructions of irreducible unitary representations Abstract: We give a review of ideas of geometric quantization and constructions of irreducuble representations. We work especially on higher-dimensional (not only on 1-dimensional, as expected) quantized G_bundle case [1]. We apply the construction to the case of some locally compact quantum groups [2], namely the quantum normalizer of SU(1,1) in SL(2,C). Selected references: [1] Do Ngoc Diep, Methods of Noncommutative Geometry for Group C*-algebras, Chapman & Hall CRC/Research Notes in mathematics Series, Vol. 416, 1999, 365pp+xx. [2] Do Ngoc Diep, Noncommutative Chern-Connes character of the locally compact quantum normalizer of SU(1,1) in SL(2,C), Intl. J. of Math., Vol. 15, No 4, 2004, 361-367. -- Tran Dao Dong: Mp^c structures and geometric quantization on U(1)-covering Abstract: TBA -- Dao Vong Duc (with Do Ngoc Diep, Ha Vinh Tan, Nguyen Ai Viet): Tetrat as key development tool of Einstein-Cartan-Evans (ECE) theory Abstract: Recently, in physics there is some astonishing event that Evans created some new unified theory for all kind of forces in physics including electromagnetical, gravitational, strong and weak etc... It is the so called new Einstein-Cartan-Evans theory. Estimated that it should be a revolutionary geometrization of physics, realizing the well-known dream of Einstein. This theory is based on differential geometry and symplectic geometry and Poisson geometry, in particular. We expose some results from Differential Geometry those are crucial for  Einstein - Cartan - Evans Theory. Our new result is the idea to consider the geometry of fiber bundles over the $(1,3)$-dimensional spacetime: we examine other models of strings as fiber bundles over the $(1,3)$-dimensional spacetime. We show that the so called ``tetrad postulate" is indeed provable.  We also prove that the string models are  fiber bundles over the $(1,3)$-dimensional spacetime. -- Rui Loja Fernandes: Symplectization commutes with reduction Abstract: To every integrable Poisson manifold $M$ one can attach a canonical symplectic object $\Sigma(M)$ (its "symplectic groupoid") which is relevant, for example, for the quantization of $M$. Given such a Poisson manifold $M$ with a free and proper action of a connected Lie group $G$ by Poisson diffeomorphisms, there is an induced free and proper Hamiltonian action of $G$ on $\Sigma(M)$. In this talk I will discuss when the equality $\Sigma(M)//G=\Sigma(M/G)$ holds and how to make sense of this in the non free case. -- Gregory Ginot: Loop product for orbifolds Abstract: Recently, motivated by physics, Chas Sullivan and al. have shown that the homology of the free loop space of a manifold has a rich algebraic structure. The more important one is a commutative associative product, called the loop product, generalizing both  intersection and Pontrjagin product. The goal of the talk is to explain how to build an analog of the Loop product for free loops and ghost loops of an orbifold. For complex orbifold, a very important such product was found by Chen and Ruan. We will explain the relationship between these two products. -- Nguyen Viet Hai: Deformation quantization and representations of some Lie groups Abstract: TBA -- Tara Holm: Orbifold cohomology of abelian symplectic reductions Abstract: I will talk about the topology of symplectic (and other) quotients. I will briefly review Kirwan's techniques for proving that the restriction map from the equivariant cohomology of the originial space to the ordinary cohomology of the symplectic reduction is a surjection. I will show how this result can be used to understand the topology of quotients, focusing on the theme of orbifolds and computing the Chen-Ruan orbifold cohomology ring of abelian symplectic reductions. -- Camille Laurent: Differential gerbes and connections, Abstract: Gerbes and connective structures on those are on object of growing interest in recent theoretical physics. We shall introduce gerbes over stacks using Lie groupoids, and say a few words about their connections. -- Aissa Wade: Remarks on deformations of generalized complex structures Abstract: TBA -- Ingo Waschkies: Simple sheaves on smooth complex Lagrangian submanifolds Following Kashiwara, quantization of complex contact manifolds can be understood as the construction of a stack of microdifferential modules. Locally, i.e. on a projective cotangent bundle associated to a complex manifold, we may look at solutions of holonomic systems and get the (purely topological) notion of microlocal perverse sheaves whose construction we will recall. In this context the quantization problem then consists in patching microlocal perverse sheaves on a complex contact manfold. The main tool in this patching process is the construction of a canonical (twisted) microlocal perverse sheaf on a smooth Lagrangian submanifold which is the topological analagon of a canonical (twisted) simple microdifferential system. In this talk, we will classify those sheaves and show how the quantization problem can be solved. -- Marco Zambon: Coisotropic embeddings in Poisson manifolds We will show that  any submanifold of a Poisson manifold $P$ satisfying a certain constant rank condition sits coisotropically inside some bigger submanifold of $P$, which is naturally endowed with a Poisson structure. Then we will give conditions under which a Dirac manifold can be embedded coisotropically in a Poisson manifold (this extends a classical theorem of Gotay). Our first result can be considered a "submanifold" version of our second result.