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This monograph gives an overview of various classes of infinite-dimensional Lie groups and their applications in Hamiltonian mechanics, fluid dynamics, integrable systems, gauge theory, and complex geometry. While infinite-dimensional groups often exhibit very peculiar features, this book describes unifying geometric ideas of the theory and gives numerous illustrations and examples, ranging from the classification of the Virasoro coadjoint orbits to knot theory, from optimal mass transport to moduli spaces of flat connections on surfaces. The text includes many exercises and open questions, and it is accessible to both students and researchers in Lie theory, geometry, and Hamiltonian systems.

Infinite dimensional Lie algebras. --- Lie groups. --- Algebra --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Groups, Lie --- Lie groups --- Grupos de --- MATEMATICAS;LIBROS ELECTRONICOS --- Mathematics. --- Algebraic geometry. --- Group theory. --- Topological groups. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Geometry. --- Physics. --- Topological Groups, Lie Groups. --- Group Theory and Generalizations. --- Mathematical Methods in Physics. --- Global Analysis and Analysis on Manifolds. --- Algebraic Geometry. --- Lie algebras --- Symmetric spaces --- Topological groups --- Topological Groups. --- Mathematical physics. --- Global analysis. --- Geometry, algebraic. --- Algebraic geometry --- Geometry --- Physical mathematics --- Physics --- Euclid's Elements --- Groups, Theory of --- Substitutions (Mathematics) --- Groups, Topological --- Continuous groups --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics

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Differential geometry. Global analysis --- Topological groups. Lie groups --- Geometry --- Mathematical physics --- Classical mechanics. Field theory --- Thermodynamics --- thermodynamica --- landmeetkunde --- topologie (wiskunde) --- differentiaal geometrie --- wiskunde --- fysica --- mechanica