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Topic: Publication of GTM volume 17 - Lectures on Poisson Geometry
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Geometry and Topology

Posts: 139
Registered: 5/24/06
Publication of GTM volume 17 - Lectures on Poisson Geometry
Posted: Apr 21, 2011 4:25 AM
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Geometry & Topology Publications is pleased to announce the publication of

Lectures on Poisson Geometry

edited by Tudor Ratiu, Alan Weinstein and Nguyen Tien Zung

Poisson geometry is a rapidly growing subject, with many interactions
and applications in areas of mathematics and physics, such as
classical differential geometry, Lie theory, noncommutative geometry,
integrable systems, fluid dynamics, quantum mechanics, and quantum
field theory. Recognizing the role played by Poisson geometry and the
significant research it has generated, the Abdus Salam International
Centre for Theoretical Physics in Trieste, Italy, sponsored a 3?week
summer activity on this subject (July 4?22 2005) in order to
bring it to the attention of scientists and students from developing
countries. There was an overwhelming response to this program, which
brought together more than 150 participants from all over the world
with varied backgrounds, from graduate students to experts.

The program consisted of a two-week intensive school comprising
10 minicourses, followed by a week-long international research
conference. The lecturers at the school were asked to turn their notes
into sections of a book that could serve as a quick introduction to
the current state of research in Poisson geometry. We hope that the
present volume will be useful to people who want to learn about
Poisson geometry and its applications.

Contents

(1) Geometry & Topology Monographs 17 (2011) 1-107
Lectures on integrability of Lie brackets
by Marius Crainic and Rui Loja Fernandes
URL: http://www.msp.warwick.ac.uk/gtm/2011/17/p001.xhtml

(2) Geometry & Topology Monographs 17 (2011) 109-169
Normal forms of Poisson structures
by Jean-Paul Dufour and Nguyen Tien Zung
URL: http://www.msp.warwick.ac.uk/gtm/2011/17/p002.xhtml

(3) Geometry & Topology Monographs 17 (2011) 171-220
Deformation quantisation of Poisson manifolds
by Simone Gutt
URL: http://www.msp.warwick.ac.uk/gtm/2011/17/p003.xhtml

(4) Geometry & Topology Monographs 17 (2011) 221-384
Applications of Poisson geometry to physical problems
by Darryl D Holm
URL: http://www.msp.warwick.ac.uk/gtm/2011/17/p004.xhtml

(5) Geometry & Topology Monographs 17 (2011) 385-472
Hamiltonian and quantum mechanics
by Anatol Odzijewicz
URL: http://www.msp.warwick.ac.uk/gtm/2011/17/p005.xhtml

(6) Geometry & Topology Monographs 17 (2011) 473-502
Lectures on Poisson groupoids
by Camille Laurent-Gengoux, Mathieu Stienon and Ping Xu
URL: http://www.msp.warwick.ac.uk/gtm/2011/17/p006.xhtml

Abstracts follow

(1) Lectures on integrability of Lie brackets
by Marius Crainic and Rui Loja Fernandes

These lectures discuss the problem of integrating
infinitesimal geometric structures to global geometric structures.


(2) Normal forms of Poisson structures
by Jean-Paul Dufour and Nguyen Tien Zung

These notes arise from a minicourse given by the two authors at the
Summer School on Poisson Geometry, ICTP, 2005. The main reference is our
recent monograph ``Poisson structures and their normal forms'', Progress
in Mathematics, Volume 242, Birkhauser, 2005. The aim of these notes
is to give an introduction to Poisson structures and a study of their
local normal forms, via Poisson cohomology and analytical techniques.


(3) Deformation quantisation of Poisson manifolds
by Simone Gutt

This introduction to deformation quantisation will focus on the
construction of star products on symplectic and Poisson manifolds. It
corresponds to the first four lectures I gave at the 2005 Summer School on
Poisson Geometry in Trieste.

The first two lectures introduced the general concept of formal
deformation quantisation with examples, with Fedosov's construction of
a star product on a symplectic manifold and with the classification of
star products on a symplectic manifold.

The next lectures introduced the notion of formality and its link with
star products, gave a flavour of Kontsevich's construction of a formality
for R^d and a sketch of the globalisation of a star product on a
Poisson manifold following the approach of Cattaneo, Felder and Tomassini.

The notes here are a brief summary of those lectures; I start with a
further reading section which includes expository papers with details
of what is presented.

In the last lectures I only briefly mentioned different aspects of the
deformation quantisation programme such as action of a Lie group on a
deformed product, reduction procedures in deformation quantisation, states
and representations in deformed algebras, convergence of deformations,
leaving out many interesting and deep aspects of the theory (such as
traces and index theorems, extension to fields theory); these are not
included in these notes and I include a bibliography with many references
to those topics.


(4) Applications of Poisson geometry to physical problems
by Darryl D Holm

These being lecture notes for a summer school, one should not seek
original material in them. Rather, the most one could hope to find
would be the insight arising from incorporating a unified approach
(based on reduction by symmetry of Hamilton's principle) with some novel
applications. I hope the reader will find insight in the lecture notes,
which are meant to be informal, more like stepping stones than a proper
path.


(5) Hamiltonian and quantum mechanics
by Anatol Odzijewicz

In these notes we review the foundations of Banach-Poisson geometry
and explain how in this framework one obtains a unified approach
to the Hamiltonian and the quantum mechanical description of the
physical systems. Our considerations will be based on the notion of
Banach Lie--Poisson space and the notion of the coherent state map,
which appear to be the crucial instrument for the clarifying what is
the quantization of the classical physical (Hamiltonian) system.


(6) Lectures on Poisson groupoids
by Camille Laurent-Gengoux, Mathieu Stienon and Ping Xu

In these lecture notes, we give a quick account of the theory of Poisson
groupoids and Lie bialgebroids. In particular, we discuss the universal
lifting theorem and its applications including integration of quasi-Lie
bialgebroids, integration of Poisson Nijenhuis structures and Alekseev
and Kosmann-Schwarzbach's theory of D/G-momentum maps.



Geometry & Topology Publications is an imprint of
Mathematical Sciences Publishers




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