Program of SMR1665
ICTP, Trieste, 04-22 July 2005
First Week (School):
Monday 04:
8h30-10h30: registration
10h30-10h45: opening (Le Dung Trang, Head of the Methamatics Section of ICTP)
10h45-12h: J.-P. Ortega, Symmetry and reduction in Poisson geometry, I.
14h-15h: J.-P. Dufour, Normal forms of Poisson structures, I.
15h20-16h20: D. Holm, Applications of Poisson geometry to physical problems, I.
16h45-18h: P. Cartier, Supersymmetric mechanics and functional integration, I
Tuesday 05:
8h45-10h: J.-P. Ortega, Symmetry and reduction in Poisson geometry, II.
10h30-12h: D. Blankenstein or T. Ratiu, Dirac structures, I.
14h-15h: J.-P. Dufour, Normal forms of Poisson structures, II.
15h20-16h20: D. Holm, Applications of Poisson geometry to physical problems, II.
16h45-18h: P. Cartier, Supersymmetric mechanics and functional integration, II.
Wednesday 06:
8h45-10h: J.-P. Ortega, Symmetry and reduction in Poisson geometry, III.
10h30-12h: D. Blankenstein or T. Ratiu, Dirac structures, II.
14h-15h: J.-P. Dufour, Normal forms of Poisson structures, III.
15h20-16h20: D. Holm, Applications of Poisson geometry to physical problems, III.
16h45-18h: P. Cartier, Supersymmetric mechanics and functional integration, III.
Thursday 07:
8h45-10h: J.-P. Ortega, Symmetry and reduction in Poisson geometry, IV.
10h30-12h: D. Blankenstein or T. Ratiu, Dirac structures, III.
14h-15h: N.T. Zung, Normal forms of Poisson structures, IV.
15h20-16h20: D. Holm, Applications of Poisson geometry to physical problems, IV.
16h45-18h: M. Crainic, Lie groupoids and Lie algebroids, I: Lie groupoids.
Friday 08:
8h45-10h: P. Cartier, Supersymmetric mechanics and functional integration, IV.
10h30-12h: D. Blankenstein or T. Ratiu, Dirac structures, IV.
14h-15h: N.T. Zung, Normal forms of Poisson structures, V.
15h20-16h20: D. Holm, Applications of Poisson geometry to physical problems, V.
16h45-18h: R. Fernandes, Lie groupoids and Lie algebroids, II: Lie algebroids.
Second Week (School):
Monday 11:
8h45-9h45: S. Gutt, Quantization of Poisson manifolds, I.
10h-11h: B. Dubrovin, Bihamiltonian structures and Frobenius manifolds, I.
11h20-12h20: A. Odzijewicz, Infinite dimensional Poisson geometry and coherent states, I.
14h30-15h45: M. Crainic, Lie groupoids and Lie algebroids, III: integrability theorem.
16h15-17h30: R. Fernandes, Lie groupoids and Lie algebroids, IV: integrability theorem.
Tuesday 12:
8h45-9h45: S. Gutt, Quantization of Poisson manifolds, II.
10h-11h: B. Dubrovin, Bihamiltonian structures and Frobenius manifolds, II.
11h20-12h20: A. Odzijewicz, Infinite dimensional Poisson geometry and coherent states, II.
14h30-15h45: P. Xu, Poisson-Lie groups and Poisson groupoids, I.
16h15-17h30: M. Crainic, Lie groupoids and Lie algebroids, V: integrability and prequantization.
Wednesday 13:
8h45-9h45: S. Gutt, Quantization of Poisson manifolds, III.
10h-11h: B. Dubrovin, Bihamiltonian structures and Frobenius manifolds, III.
11h20-12h20: A. Odzijewicz, Infinite dimensional Poisson geometry and coherent states, III.
14h30-15h45: P. Xu, Poisson-Lie groups and Poisson groupoids, II.
16h15-17h30: M. Crainic, Lie groupoids and Lie algebroids, VI: integrability and cohomology.
Thursday 14:
8h45-9h45: S. Gutt, Quantization of Poisson manifolds, IV.
10h-11h: B. Dubrovin, Bihamiltonian structures and Frobenius manifolds, IV.
11h20-12h20: A. Odzijewicz, Infinite dimensional Poisson geometry and coherent states, IV.
14h30-15h45: P. Xu, Poisson-Lie groups and Poisson groupoids, III.
16h15-17h30: R. Fernandes, Lie groupoids and Lie algebroids, VII: integrability and
Poisson geometry.
Friday 15:
8h45-9h45: S. Gutt, Quantization of Poisson manifolds, V.
10h-11h: B. Dubrovin, Bihamiltonian structures and Frobenius manifolds, V.
11h20-12h20: A. Odzijewicz, Infinite dimensional Poisson geometry and coherent states, V.
14h30-15h45: P. Xu, Poisson-Lie groups and Poisson groupoids, IV.
16h15-17h30: R. Fernandes, Lie groupoids and Lie algebroids, VIII: integrability,
foliation theory and homotopy theory.
Saturday 16: excursion to Grotta Gigante.
Third Week (Conference):
Monday 18:
10h-10h10: opening (Alan Weinstein)
10h10-11h05: V. Guillemin, The notion of moment map for families of symplectomorphisms. (abstract, slides)
11h30-12h25: J.H. Lu, Examples of Poisson varieties associated to semi-simple groups. (abstract, slides)
14h30-15h25: A. Alekseev, Ginzburg-Weinstein via Gelfand-Zeitlin. (abstract, slides)
15h40-16h35: M. Boucetta, On the Riemann-Poisson manifolds. (abstract, slides)
17h-17h55: C. Laurent, Connections of non-abelian differential gerbes. (abstract, slides)
Tuesday 19:
9h-9h55: N. Poncin, Automorphisms and derivations of quantum and classical Poisson algebras. (abstract, slides)
10h10-11h05: M. Karasev, Resonances and quantized space-time. (abstract, slides)
11h30-12h25: Y. Maeda, Deformation quantizations and gerbes. (abstract, slides)
14h30-15h25: Y. Kosmann-Schwarzbach, The modular class in Lie algebroid theory. (abstract, slides)
15h40-16h35: V. Ginzburg, Morse theory and algebras of moduli spaces. (abstract, slides)
17h-17h55: H. Bursztyn, Quasi-Poisson theory via Dirac geometry. (abstract, slides)
Wednesday 20:
9h-9h55: T. Ratiu, Reduction for general symplectic actions. (abstract, slides)
10h10-11h05: A. Weinstein, Poisson geometry and deformation quantization near a pseudoconvex boundary. (abstract, slides)
11h30-12h25: S. Gutt, Ricci-flat symplectic connections. (abstract, slides)
Free afternoon (excursion to Duino castle)
Thursday 21:
9h-9h55: M. Crainic, On multiplicative 2-forms. (abstract, slides)
10h10-11h05: N. Hitchin, Poisson structures and bihermitian metrics. (abstract, slides)
11h30-12h25: A. Odzijewicz, Noncommutative Kahler-like structures in quantization. (abstract, slides)
14h30-15h25: P. Xu, Some remarks on generalized complex structures. (abstract, slides)
15h50-16h45: A. Bolsinov and O. Yakomivo, Complete commutative subalgebras in Poisson-Lie algebras:
a proof of the Mischenko-Fomenko conjecture and generalizations. (abstract, slides)
Friday 22:
9h-9h55: Z.J. Liu, On (co-)morphisms of Lie pseudoalgebras and groupoids. (abstract, slides)
10h10-11h05: V. Fock, Poisson structure and quantization of cluster varieties. (abstract, slides)
11h30-12h25: M. Schlinchenmaier, Berezin-Toeplitz quantization of the moduli space of flat SU(N) connections. (abstract, slides)
14h30-15h25: Yu. Vorobjev, Symplectic leaf neighborhood theorems. (abstract, slides)
15h50-16h45: A. Wade, Dirac structures and the phase space of particles in a Yang-Mills-Higgs field. (abstract, slides)
(closing)
Some Abstracts:
A. Alekseev (University of Geneva)
Title: Ginzburg-Weinstein via Gelfand-Zeitlin
Abstract:
Let U(n) be the unitary group, and $u(n)^*$ the dual of its Lie
algebra, equipped with the Kirillov Poisson structure. In their 1983
paper, Guillemin-Sternberg introduced a densely defined Hamiltonian
action of a torus of dimension $(n-1)n/2$ on $u(n)^*$, with moment map
given by the Gelfand-Zeitlin coordinates. A few years later,
Flaschka-Ratiu described a similar, `multiplicative' Gelfand-Zeitlin
system for the Poisson Lie group $U(n)^*$. By the Ginzburg-Weinstein
theorem, $U(n)^*$ is isomorphic to $u(n)^*$ as a Poisson manifold.
Flaschka-Ratiu conjectured that one can choose the Ginzburg-Weinstein
diffeomorphism in such a way that it intertwines the
linear and nonlinear Gelfand-Zeitlin systems. Our main result gives a
proof of this conjecture, and produces a canonical Ginzburg-Weinstein
diffeomorphism.
The talk is based on the joint work with E. Meinrenken, math.DG/0506112.
M. Boucetta
"On the Riemann-Poisson manifolds".
Abstract. A Riemann-Poisson manifold is a Riemannian manifold endowed
with a Poisson tensor parallel with respect to the contravariant
Levi-Civita connection associated with the metric and the Poisson
tensor. The aim of this talk is to introduce the notion of
Riemann-Poisson manifold, to give some general properties of such
structures. Many examples will be given in order to show the interest
of this notion.
Vladimir Fock
Poisson structure and quantization of cluster varieties.
Abstract:
Cluster varieties is a class of varieties admitting certain discrete
set of birational parameterizations by algebraic tori with prescribed
transition functions. Among such varieties there are Lie groups,
grassmanians, character varieties of 2D Riemann surfaces, Teichmueller
spaces, certain configuration spaces and many others. Cluster approach
to these varieties allows to describe explicitly and in a unified way
such structures as the canonical Poisson structure as well as its
quantization, canonical discrete group action respecting all these
structure, tropicalisation, canonical bases of functions and many
others.
The talk is based on joint work with A.B.Goncharov.
Victor Guillemin
Title: The notion of moment map for families of symplectomorphisms.
Abstract: As Alan Weinstein observed in his article "Symplectic
geometry" (Bull. AMS 1981, 1-13), the moment map associated with the
Hamiltonian action of a Lie group defines a "moment Lagrangian" whose
properties encode many of the main features of moment geometry. In my
talk I'll describe an analogous object for families of
symplectomorphisms (i.e. no groups.)
ZJ Liu
title: On (co-)morphisms of Lie pseudoalgebras and groupoids (joint work with Z. Chen)
Abstract: A unified description of morphisms and comorphisms of Lie
pseudoalgebras is given by showing that both of their graphs can be
realized as Lie subpseudoalgebras of a Lie pseudoalgebra, which is
named the $\psi$ sum defined in this paper. We also provide similar
descriptions for the morphisms and the comorphisms of Lie algebroids
and groupoids. It will be seen that the three versions of our main
theorems express actually a same fact from different points of view
(algebraic, geometric and global versions respectively).
Mikhail Karasev
Resonances and Quantized Space-Time
We observe that in wave mechanics, in nano- and micro-zones near
resonance equilibria, or resonance trajectories, or resonance tori,
the dynamic algebras turn out to be noncommutative. The commutation
relations between generators of these algebras are determined by the
arithmetic type of resonance.
The simplest example: the Landau levels of a charge in a magnetic field
(neutral resonance, nilpotent dynamic algebra), the related quantum Hall
effect and noncommutativity of the configuration plane.
We consider, in more details, the case of general elliptic resonance. Its
dynamic algebra is of compact type. For two degrees of freedom, we
present the polynomial Poisson and quantum tensors of this algebra.
The quantum Kahlerian structure on symplectic leaves, the reproducing
measure, the coherent states and Hermitian irreducible representations
are explicitly derived using hypergeometric functions.
Examples: the noncommutative nano-electrodynamics in resonance
optical channels (via the Helmholtz equation), the so-called "quantum
dots", two dimensional magneto-atoms, etc.
We demonstrate that the noncommutativity of the configuration space
in the elliptic case is induced by a certain triple brackets structure. The
resonance precession system is Hamiltonian in noncommutative space-
time coordinates.
Jiang-Hua Lu
Title: Examples of Poisson varieties associated to semi-simple groups
Abstract: We will describe a class of Poisson varieties with the
actions by two groups $G_1$ and $G_2$ such that all non-empty
intersections of $G_1$ and $G_2$ orbits are regular Poisson
subvarieties. This is joint work with M. Yakimov.
Yoshiaki Maeda
"Deformation quantizations and gerbes"
Abstract: We will present how an example of gerbes appears naturally
from the deformation quantization. We will also give some basic
examples of gerbes which seems toy models of our construction of the
gerbes of star exponential functions of quadratics in the Weyl algebra.
We also will propose a new geometric object, which we would like to
call "pile" , by introducing flat connections on the gerbes.
A. Odzijewicz
Noncommutative Kahler-like structures in quantization
Abstract:
There are complementary methods of mathematical description od quantum
physical systems. These are the algebraicmethods based on the theory of
$C^*$-algebras and the geometric methods that
find an elegant
presentation as Kostant-Souriau quantization and $*-$product quantization.
In our approach we take an effort to unify both these methods by use of the
notion of coherent states
map. The coherent states map $\K$ means a symplectic map of the classical phase
space $M$
into quantum phase space, i.e. complex projective Hilbert space $\CP(\M)$. It
appears that using
the coherent states map one can unify, in some sense, the classical and quantum
description of the
considered physical system.
Bearing in mind the above fact we introduce the notion of a polarized algebra of
observables $\A$,
which is univocally determined by coherent states map $\K$. In the case when
$\K$ is Gaussian coherent states map of
linear phase space $\R^{2N}$ into $\CP(\M)$, the algebra $\A$ is Heisenberg-Weyl
algebra. So the $C^*$-algebra $\A$ is
a natural generalization of the latter one to the case of a general phase space
$M$. We prove some important properties
of $\A$ and explain the relation of the structure of $\A$ to the structures such
as e.g. prequantum line bundle and
polarization which play a crucial role in Kostant-Souriau quantization.
As a result one can distinguish additional structures on $C^*$-algebra $\A$ that
are responsible for the symplectic
(K\"ahler in the special case) structures of classical phase space $M$. This
structure denotes the existence of a
commutative Banach subalgebra $\overline\P$ of $\A$ which has physical
interpretation as the algebra of annihilation
operators. Its classical counterpart is the polarization in the sense of
Kostant-Souriau quantization. So, it is
natural to understand $\overline\P$ as the quantum polarization and call
$(\A,\overline\P)$ the polarized
$C^*$-algebras or quantum K\"ahler manifold.
We introduce the notion of an abstract coherent state on $(\A,\overline\P)$. The
coherent states in this sense generalize the notion of vacuum to the case of a
general phase space.
On the another hand using the coherent states
one can study the algebra $\A$ by reducing many problems to the investigation of
its polarization
$\overline\P$, which is more handy because of commutativity.
Also we show the fundamental properties of coherent states: on $\A$ they can be
considered
as classical states of some classical phase space being subspace of space of
multiplicative
functional on the polarization $\overline\P$. Additionally, when we apply the
GNS construction to the
coherent states we obtain Hilbert space which is a generalization of Hardy
space, which is exactly
obtained when $M=\D$ is a unit disc in $\C$ and $\A$ is Toeplitz algebra.
We show how to reconstruct from $(\A,\overline\P)$ the classical phase space
$M$ and coherent states map $\K$. This gives rise to the method of
reconstruction of the classical
mechanics picture for the quantum one.
Norbert Poncin
Automorphisms and derivations of quantum and classical Poisson algebras
Refs:
Automorphisms of quantum and classical Poisson algebras
http://arxiv.org/abs/math.RA/0211175
Derivations of the Lie Algebras of Differential Operators
http://arxiv.org/abs/math.DG/0312162
Lie algebraic characterization of manifolds
http://arxiv.org/abs/math.DG/0310202
Equivariant symbol calculus for differential operators acting on forms
http://arxiv.org/abs/math.RT/0206213
Abstract:
Algebraic characterizations of topological spaces and manifolds go back
to the Gelfand-Neumark theory. The classical result of Pursell and
Shanks, which states that the Lie algebra of smooth vector fields over
a smooth manifold characterizes the smooth structure of the variety, is
the starting point of a multitude of papers. Many types of Lie algebras
of vector fields have been considered. Let us mention the Lie algebra
of vector fields preserving a given generalized foliation, the Lie
algebras of Hamiltonian vector fields or Poisson brackets of functions,
of Jacobi brackets in general, Lie algebras of vector fields on orbit
spaces and $G$-manifolds, Lie algebras of vector fields on affine and
toric varieties, Lie algebroids, ...
The first objective of our joint works with J. Grabowski is to prove
that the Lie algebra ${\cal D}(M)$ of all linear differential
operators, its Lie-subalgebra ${\cal D}^1(M)$ of all first order
differential operators, and the Poisson-Lie algebra ${\cal S}(M)$ of
all symmetric contravariant tensors (the symbols of the preceding
operators) over a smooth, a real-analytic or holomorphic manifold $M$,
characterize the structure of $M$.
Our algebraic approach to these problems leads to the definition of
quantum Poisson algebras and classical Poisson algebras. We show that
if two (quantum or classical) Poisson algebras are isomorphic as Lie
algebras, their "basic algebras of functions" are isomorphic as
associative algebras---an algebraic Shanks-Pursell type result, which
implies the aforementioned initial goal.
We also provide an explicit description of all automorphisms and
derivations of the infinite-dimensional algebras ${\cal D}^1(M)$,
${\cal S}(M)$, and ${\cal D}(M)$. The problem of distinguishing those
derivations that generate one-parameter groups of automorphisms and
describing these groups will also be solved.
Parts of our proofs are based upon canonical and equivariant
quantization. We will briefly present these techniques and compare
their efficiency as computing devices.
Martin Schlichenmaier, University of Luxembourg
Berezin-Toeplitz quantization of the moduli space of flat SU(N) connections
As was shown by Bordemann, Meinrenken, and Schlichenmaier the Berezin-Toeplitz
operator quantization and its associated star product give a unique natural quantization
for a quantisable compact Kahler manifold. This procedure is applied for the moduli space
of gauge equivalence classes of SU(N) connections on a fixed Riemann surface. In this
context the Verlinde spaces and the Verlinde bundle over Teichmuller space show up. As
it is well-known these moduli spaces can also be described as the moduli spaces of stable
rang N algebraic bundles. Recent results of J. Andersen on the asymptotic faithfulness of
the representation of the mapping class group on the space of covariantly constant sections
of the Verlinde bundle are presented.
Yuri Vorobiev
Title: " Symplectic Leaf Neighborhood Theorems "
Abstract: We discuss some criteria of Poisson equivalence over a symplectic leaf
which are based on a homotopy argument for coupling Poisson structures. We also
derive necessary and sufficient conditions for the linearizability of a Poisson
structure at a (singular) symplectic leaf in the case when the transverse Lie
algebra of the leaf is semisimple of compact type. This semilocal generalization
of Conn's Linearization Theorem shows that, in general, there is a cohomological
obstruction to the Poisson linearizability at a symplectic leaf of nonzero dimension.
Alan Weinstein
Poisson geometry and deformation quantization near a pseudoconvex boundary
(Joint work with Eric Leichtnam and Xiang Tang)
Let $X$ be a complex manifold with strongly pseudoconvex boundary $M$.
If $\psi$ is a defining function for $M$, then $-\log\psi$ is
plurisubharmonic in a neighborhood of $X$, and the 2-form $\sigma = i
\del \delbar(-\log \psi)$ is the symplectic form associated to a
K\"ahler structure on the complement of $M$ in a neighborhood in $X$
of $M$; it blows up along $M$.
We show that the Poisson structure obtained by inverting $\sigma$
extends smoothly across $M$ and that, up to isomorphism near $M$, it
is completely determined by the contact structure on $M$ when $M$ is
compact. We also study the boundary behavior of the
``Berezin-Toeplitz" deformation quantization attached to the K\"ahler
structure. The proofs use a complex Lie algebroid determined by the
CR structure on $M$, along with some ideas of Epstein, Melrose, and
Mendoza concerning manifolds with contact boundary.
P. Xu
Some remarks on generalized complex structures
We discuss two aspects of generalized complex structures from the viewpoint
of Poisson geometry.
(1) We study the reduction of generalized complex structures generalizing
the Marsden-Weinstein reduction in symplectic case and the complex structure
reduction of a Kahler manifold as studied by Guillemin-Sternberg, and
Hitchin et. al.
(2) We study the relation between generalized complex structures and
Poisson-Nijenhuis structures.
This is a joint work with Mathieu Stienon.