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The aim of this monograph is to give an overview of various classes
of in?ni- dimensional Lie groups and their applications, mostly in
Hamiltonian - chanics, ?uid dynamics, integrable systems, and
complex geometry. We have chosen to present the unifying ideas of
the theory by concentrating on speci?c
typesandexamplesofin?nite-dimensionalLiegroups. Ofcourse,
theselection of the topics is largely in?uenced by the taste of the
authors, but we hope
thatthisselectioniswideenoughtodescribevariousphenomenaarisinginthe
geometry of in?nite-dimensional Lie groups and to convince the
reader that they are appealing objects to study from both purely
mathematical and more applied points of view. This book can be
thought of as complementary to the existing more algebraic
treatments, in particular, those covering the str- ture and
representation theory of in?nite-dimensional Lie algebras, as well
as to more analytic ones developing calculus on in?nite-dimensional
manifolds. This monograph originated from advanced graduate courses
and mi- courses on in?nite-dimensional groups and gauge theory
given by the ?rst author at the University of Toronto, at the CIRM
in Marseille, and at the Ecole Polytechnique in Paris in 2001-2004.
It is based on various classical and
recentresultsthathaveshapedthisnewlyemergedpartofin?nite-dimensional
geometry and group theory. Our intention was to make the book
concise, relatively self-contained, and useful in a graduate
course. For this reason, throughout the text, we have included a
large number of problems, ranging from simple exercises to open
questions
The aim of this monograph is to give an overview of various classes
of in?ni- dimensional Lie groups and their applications, mostly in
Hamiltonian - chanics, ?uid dynamics, integrable systems, and
complex geometry. We have chosen to present the unifying ideas of
the theory by concentrating on speci?c
typesandexamplesofin?nite-dimensionalLiegroups. Ofcourse,
theselection of the topics is largely in?uenced by the taste of the
authors, but we hope
thatthisselectioniswideenoughtodescribevariousphenomenaarisinginthe
geometry of in?nite-dimensional Lie groups and to convince the
reader that they are appealing objects to study from both purely
mathematical and more applied points of view. This book can be
thought of as complementary to the existing more algebraic
treatments, in particular, those covering the str- ture and
representation theory of in?nite-dimensional Lie algebras, as well
as to more analytic ones developing calculus on in?nite-dimensional
manifolds. This monograph originated from advanced graduate courses
and mi- courses on in?nite-dimensional groups and gauge theory
given by the ?rst author at the University of Toronto, at the CIRM
in Marseille, and at the Ecole Polytechnique in Paris in 2001-2004.
It is based on various classical and
recentresultsthathaveshapedthisnewlyemergedpartofin?nite-dimensional
geometry and group theory. Our intention was to make the book
concise, relatively self-contained, and useful in a graduate
course. For this reason, throughout the text, we have included a
large number of problems, ranging from simple exercises to open
questions
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