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.
