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.
(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