GAP4: Aspects of Quantization
Preliminary Program
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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.