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Poisson structures appear in a large variety of contexts, ranging
from string theory, classical/quantum mechanics and differential
geometry to abstract algebra, algebraic geometry and representation
theory. In each one of these contexts, it turns out that the
Poisson structure is not a theoretical artifact, but a key element
which, unsolicited, comes along with the problem that is
investigated, and its delicate properties are decisive for the
solution to the problem in nearly all cases. Poisson Structures is
the first book that offers a comprehensive introduction to the
theory, as well as an overview of the different aspects of Poisson
structures. The first part covers solid foundations, the central
part consists of a detailed exposition of the different known types
of Poisson structures and of the (usually mathematical) contexts in
which they appear, and the final part is devoted to the two main
applications of Poisson structures (integrable systems and
deformation quantization). The clear structure of the book makes it
adequate for readers who come across Poisson structures in their
research or for graduate students or advanced researchers who are
interested in an introduction to the many facets and applications
of Poisson structures.
In the early 70's and 80's the field of integrable systems was in
its prime youth: results and ideas were mushrooming all over the
world. It was during the roaring 70's and 80's that a first version
of the book was born, based on our research and on lectures which
each of us had given. We owe many ideas to our colleagues Teruhisa
Matsusaka and David Mumford, and to our inspiring graduate students
(Constantin Bechlivanidis, Luc Haine, Ahmed Lesfari, Andrew
McDaniel, Luis Piovan and Pol Vanhaecke). As it stood, our first
version lacked rigor and precision, was rough, dis connected and
incomplete. . . In the early 90's new problems appeared on the
horizon and the project came to a complete standstill, ultimately
con fined to a floppy. A few years ago, under the impulse of Pol
Vanhaecke, the project was revived and gained real momentum due to
his insight, vision and determination. The leap from the old to the
new version is gigantic. The book is designed as a teaching
textbook and is aimed at a wide read ership of mathematicians and
physicists, graduate students and professionals."
In the early 70's and 80's the field of integrable systems was in
its prime youth: results and ideas were mushrooming all over the
world. It was during the roaring 70's and 80's that a first version
of the book was born, based on our research and on lectures which
each of us had given. We owe many ideas to our colleagues Teruhisa
Matsusaka and David Mumford, and to our inspiring graduate students
(Constantin Bechlivanidis, Luc Haine, Ahmed Lesfari, Andrew
McDaniel, Luis Piovan and Pol Vanhaecke). As it stood, our first
version lacked rigor and precision, was rough, dis connected and
incomplete. . . In the early 90's new problems appeared on the
horizon and the project came to a complete standstill, ultimately
con fined to a floppy. A few years ago, under the impulse of Pol
Vanhaecke, the project was revived and gained real momentum due to
his insight, vision and determination. The leap from the old to the
new version is gigantic. The book is designed as a teaching
textbook and is aimed at a wide read ership of mathematicians and
physicists, graduate students and professionals."
This book treats the general theory of Poisson structures and integrable systems on affine varieties in a systematic way. Special attention is drawn to algebraic completely integrable systems. Several integrable systems are constructed and studied in detail and a few applications of integrable systems to algebraic geometry are worked out.In the second edition some of the concepts in Poisson geometry are clarified by introducting Poisson cohomology; the Mumford systems are constructed from the algebra of pseudo-differential operators, which clarifies their origin; a new explanation of the multi Hamiltonian structure of the Mumford systems is given by using the loop algebra of sl(2); and finally Goedesic flow on SO(4) is added to illustrate the linearizatin algorith and to give another application of integrable systems to algebraic geometry.
Poisson structures appear in a large variety of contexts, ranging
from string theory, classical/quantum mechanics and differential
geometry to abstract algebra, algebraic geometry and representation
theory. In each one of these contexts, it turns out that the
Poisson structure is not a theoretical artifact, but a key element
which, unsolicited, comes along with the problem that is
investigated, and its delicate properties are decisive for the
solution to the problem in nearly all cases. Poisson Structures is
the first book that offers a comprehensive introduction to the
theory, as well as an overview of the different aspects of Poisson
structures. The first part covers solid foundations, the central
part consists of a detailed exposition of the different known types
of Poisson structures and of the (usually mathematical) contexts in
which they appear, and the final part is devoted to the two main
applications of Poisson structures (integrable systems and
deformation quantization). The clear structure of the book makes it
adequate for readers who come across Poisson structures in their
research or for graduate students or advanced researchers who are
interested in an introduction to the many facets and applications
of Poisson structures.
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