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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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First published in 1998 this unique monograph treats topological, group-theoretic, and geometric problems of ideal hydrodynamics and magneto-hydrodynamics from a unified point of view. It describes the necessary preliminary notions both in hydrodynamics and pure mathematics with numerous examples and figures. This book, now accepted as one of the main references in the field, is accessible to graduates as well as pure and applied mathematicians working in hydrodynamics, Lie groups, dynamical systems, and differential geometry. The updated second edition also contains a survey of recent developments in this now-flourishing field of topological and geometric hydrodynamics.

Mathematics. --- Fluid mechanics. --- System theory. --- Computational intelligence. --- Mathematical physics. --- Dynamics. --- Ergodic theory. --- Mathematics, general. --- Engineering Fluid Dynamics. --- Complex Systems. --- Computational Intelligence. --- Theoretical, Mathematical and Computational Physics. --- Dynamical Systems and Ergodic Theory. --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Physical mathematics --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- Systems, Theory of --- Systems science --- Science --- Hydromechanics --- Continuum mechanics --- Math --- Philosophy --- Hydrodynamics. --- Fluid dynamics --- Hidrodinàmica --- Topologia --- Poliedres --- Teoria de conjunts --- Aplicacions (Matemàtica) --- Categories (Matemàtica) --- Compactificacions --- Conjunts de Borel --- Dinàmica topològica --- Espais mètrics --- Espais vectorials topològics --- Grups de transformacions --- Grups topològics --- Jocs d'estratègia (Matemàtica) --- Politops --- Teoria de l'homotopia --- Teoria de la dimensió (Topologia) --- Teoria de la dualitat (Matemàtica) --- Teoria de grafs --- Teoria dels reticles --- Topologia algebraica --- Topologia combinatòria --- Topologia diferencial --- Varietats (Matemàtica) --- Varietats topològiques --- Xarxes (Matemàtica) --- Àlgebra lineal --- Dinàmica de fluids --- Enginyeria hidràulica --- Mecànica --- Mecànica analítica --- Mecànica de fluids --- Cavitació --- Ones --- Ones de gravetat --- Turbulència --- Urodinàmica --- Viscositat --- Vòrtexs

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Discrete mathematics --- Mathematics --- Classical mechanics. Field theory --- Fluid mechanics --- Statistical physics --- Artificial intelligence. Robotics. Simulation. Graphics --- neuronale netwerken --- fuzzy logic --- cybernetica --- grafentheorie --- statistiek --- systeemtheorie --- wiskunde --- KI (kunstmatige intelligentie) --- ingenieurswetenschappen --- fysica --- dynamica --- vloeistoffen

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The volume is a collection of refereed research papers on infinite dimensional groups and manifolds in mathematics and quantum physics. Topics covered are: new classes of Lie groups of mappings, the Burgers equation, the Chern--Weil construction in infinite dimensions, the Hamiltonian approach to quantum field theory, and different aspects of large N limits ranging from approximation methods in quantum mechanics to modular forms and string/gauge theory duality. Directed at research mathematicians and theoretical physicists as well as graduate students, the volume gives an overview of important themes of research at the forefront of mathematics and theoretical physics.

Infinite-dimensional manifolds. --- Infinite dimensional Lie algebras. --- Differential equations, Partial. --- Quantum field theory. --- Relativistic quantum field theory --- Field theory (Physics) --- Quantum theory --- Relativity (Physics) --- Lie algebras --- Manifolds, Infinite-dimensional --- Global analysis (Mathematics) --- Topological manifolds --- Partial differential equations

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Algebra --- Partial differential equations --- Mathematical analysis --- Mathematical physics --- differentiaalvergelijkingen --- algebra --- analyse (wiskunde) --- theoretische fysica