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

Vladimir Igorevich Arnold is one of the most influential mathematicians of our time. V. I. Arnold launched several mathematical domains (such as modern geometric mechanics, symplectic topology, and topological fluid dynamics) and contributed, in a fundamental way, to the foundations and methods in many subjects, from ordinary differential equations and celestial mechanics to singularity theory and real algebraic geometry. Even a quick look at a partial list of notions named after Arnold already gives an overview of the variety of such theories and domains: KAM (Kolmogorov-Arnold-Moser) theory, The Arnold conjectures in symplectic topology, The Hilbert-Arnold problem for the number of zeros of abelian integrals, Arnold's inequality, comparison, and complexification method in real algebraic geometry, Arnold-Kolmogorov solution of Hilbert's 13th problem, Arnold's spectral sequence in singularity theory, Arnold diffusion, The Euler-Poincare-Arnold equations for geodesics on Lie groups, Arnold's stability criterion in hydrodynamics, ABC (Arnold-Beltrami-Childress) ?ows in ?uid dynamics, The Arnold-Korkina dynamo, Arnold's cat map, The Arnold-Liouville theorem in integrable systems, Arnold's continued fractions, Arnold's interpretation of the Maslov index, Arnold's relation in cohomology of braid groups, Arnold tongues in bifurcation theory, The Jordan-Arnold normal forms for families of matrices, The Arnold invariants of plane curves. Arnold wrote some 700 papers, and many books, including 10 university textbooks. He is known for his lucid writing style, which combines mathematical rigour with physical and geometric intuition. Arnold's books on Ordinarydifferentialequations and Mathematical methodsofclassicalmechanics became mathematical bestsellers and integral parts of the mathematical education of students throughout the world.

This newly-translated book takes the reader from the basic principles and conservation laws of hydrodynamics to the description of general atmospheric circulation. Among the topics covered are the Kelvin, Ertel and Rossby-Obukhov invariants, quasi-geostrophic equation, thermal wind, singular Helmholtz vortices, derivation of the Navier-Stokes equation, Kolmogorov's flow, hydrodynamic stability, and geophysical boundary layers. Generalizing V. Arnold's approach to hydrodynamics, the author ingeniously brings in an analogy of Coriolis forces acting on fluid with motion of the Euler heavy top and shows how this is used in the analysis of general atmospheric circulation.

This book is based on popular graduate and undergraduate courses given by F.V.Dolzhansky at the Moscow Institute of Physics and Technology, and is the result of the author's highly acclaimed work in Moscow's Laboratory of Geophysical Hydrodynamics. Each chapter is full of examples and figures, exercises and hints, motivating and illustrating both theoretical and experimental results. The exposition is comprehensive yet user-friendly in engaging and exploring the broad range of topics for students and researchers in mathematics, physics, meteorology and engineering. "

Volume III of the Collected Works of V.I. Arnold contains papers written in the years 1972 to 1979. The main theme emerging in Arnold's work of this period is the development of singularity theory of smooth functions and mappings. The volume also contains papers by V.I. Arnold on catastrophe theory and on A.N. Kolmogorov's school, his prefaces to Russian editions of several books related to singularity theory, V. Arnold's lectures on bifurcations of discrete dynamical systems, as well as a review by V.I. Arnold and Ya.B. Zeldovich of V.V. Beletsky's book on celestial mechanics. Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors.

The subject of this book is the methodology and results of integrated geophysical investigations in the Caucasian region, mainly interpretation of magnetic and gravity anomalies with utilization of a huge petrophysical database for the evaluation of geological structure and mineral resources. Relative voluminous geophysical data are useful for the Earth Sciences researchers interested in the Caucasian region (and adjacent and similar regions) characterized by complicated geological structure, inclined magnetization (polarization), uneven topography and mountain/sea transition. Examination of geophysical fields verified by super-deep wells drilling indicates that magmatic rocks of the Lesser Caucasus are extended northward under thick sedimentary cover of the Kura Depression up to the Greater Caucasus. These rocks form hidden petroleum-bearing traps of a newly identified type. On the basis of geophysical studies (mainly inexpensive magnetic and electric methods), a new copper-polymetallic province in the Greater Caucasus has been revealed. a newly developed integrated approach and special information-statistical techniques for processing and interpretation of geophysical data facilitate detection of important geological features, e.g. hidden intersections of linear structures that control location of large commercial ore and oil-and-gas deposits, as well as focuses of dangerous geodynamic events at a depth. Numerous illustrations (including colour) elucidate different problems and solutions on various scales and in diverse geological-geophysical environments. Many aspects of this book have been presented at the teaching courses for bachelors, masters and doctors at the Tel-AvivUniversity (Tel Aviv, Israel) and Ben-Gurion University (Bea (TM)er-Sheva, Israel). Benefits to readers are predetermined by the combination of the authors many-years personal experience in the geophysical studies of Azerbaijan and other regions of the Caucasus with the authora (TM)s knowledge of the modern level of geophysics in the world.

Volume III of the Collected Works of V.I. Arnold contains papers written in the years 1972 to 1979. The main theme emerging in Arnold's work of this period is the development of singularity theory of smooth functions and mappings. The volume also contains papers by V.I. Arnold on catastrophe theory and on A.N. Kolmogorov's school, his prefaces to Russian editions of several books related to singularity theory, V. Arnold's lectures on bifurcations of discrete dynamical systems, as well as a review by V.I. Arnold and Ya.B. Zeldovich of V.V. Beletsky's book on celestial mechanics. Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors.

This newly-translated book takes the reader from the basic principles and conservation laws of hydrodynamics to the description of general atmospheric circulation. Among the topics covered are the Kelvin, Ertel and Rossby-Obukhov invariants, quasi-geostrophic equation, thermal wind, singular Helmholtz vortices, derivation of the Navier-Stokes equation, Kolmogorov's flow, hydrodynamic stability, and geophysical boundary layers. Generalizing V. Arnold's approach to hydrodynamics, the author ingeniously brings in an analogy of Coriolis forces acting on fluid with motion of the Euler heavy top and shows how this is used in the analysis of general atmospheric circulation. This book is based on popular graduate and undergraduate courses given by F.V.Dolzhansky at the Moscow Institute of Physics and Technology, and is the result of the author's highly acclaimed work in Moscow's Laboratory of Geophysical Hydrodynamics. Each chapter is full of examples and figures, exercises and hints, motivating and illustrating both theoretical and experimental results. The exposition is comprehensive yet user-friendly in engaging and exploring the broad range of topics for students and researchers in mathematics, physics, meteorology and engineering.

The subject of this book is the methodology and results of integrated geophysical investigations in the Caucasian region, mainly interpretation of magnetic and gravity anomalies with utilization of a huge petrophysical database for the evaluation of geological structure and mineral resources. Relative voluminous geophysical data are useful for the Earth Sciences researchers interested in the Caucasian region (and adjacent and similar regions) characterized by complicated geological structure, inclined magnetization (polarization), uneven topography and mountain/sea transition. Examination of geophysical fields verified by super-deep wells drilling indicates that magmatic rocks of the Lesser Caucasus are extended northward under thick sedimentary cover of the Kura Depression up to the Greater Caucasus. These rocks form hidden petroleum-bearing traps of a newly identified type. On the basis of geophysical studies (mainly inexpensive magnetic and electric methods), a new copper-polymetallic province in the Greater Caucasus has been revealed. a newly developed integrated approach and special information-statistical techniques for processing and interpretation of geophysical data facilitate detection of important geological features, e.g. hidden intersections of linear structures that control location of large commercial ore and oil-and-gas deposits, as well as focuses of dangerous geodynamic events at a depth. Numerous illustrations (including colour) elucidate different problems and solutions on various scales and in diverse geological-geophysical environments. Many aspects of this book have been presented at the teaching courses for bachelors, masters and doctors at the Tel-Aviv University (Tel Aviv, Israel) and Ben-Gurion University (Be'er-Sheva, Israel). Benefits to readers are predetermined by the combination of the authors many-years personal experience in the geophysical studies of Azerbaijan and other regions of the Caucasus with the authors' knowledge of the modern level of geophysics in the world.

Vladimir Igorevich Arnold is one of the most influential mathematicians of our time. V. I. Arnold launched several mathematical domains (such as modern geometric mechanics, symplectic topology, and topological fluid dynamics) and contributed, in a fundamental way, to the foundations and methods in many subjects, from ordinary differential equations and celestial mechanics to singularity theory and real algebraic geometry. Even a quick look at a partial list of notions named after Arnold already gives an overview of the variety of such theories and domains: KAM (Kolmogorov-Arnold-Moser) theory, The Arnold conjectures in symplectic topology, The Hilbert-Arnold problem for the number of zeros of abelian integrals, Arnold's inequality, comparison, and complexification method in real algebraic geometry, Arnold-Kolmogorov solution of Hilbert's 13th problem, Arnold's spectral sequence in singularity theory, Arnold diffusion, The Euler-Poincare-Arnold equations for geodesics on Lie groups, Arnold's stability criterion in hydrodynamics, ABC (Arnold-Beltrami-Childress) ?ows in ?uid dynamics, The Arnold-Korkina dynamo, Arnold's cat map, The Arnold-Liouville theorem in integrable systems, Arnold's continued fractions, Arnold's interpretation of the Maslov index, Arnold's relation in cohomology of braid groups, Arnold tongues in bifurcation theory, The Jordan-Arnold normal forms for families of matrices, The Arnold invariants of plane curves. Arnold wrote some 700 papers, and many books, including 10 university textbooks. He is known for his lucid writing style, which combines mathematical rigour with physical and geometric intuition. Arnold's books on Ordinarydifferentialequations and Mathematical methodsofclassicalmechanics became mathematical bestsellers and integral parts of the mathematical education of students throughout the world."

Helmholtz's seminal paper on vortex motion (1858) marks the beginning of what is now called topological fluid mechanics.After 150 years of work, the field has grown considerably. In the last several decades unexpected developments have given topological fluid mechanics new impetus, benefiting from the impressive progress in knot theory and geometric topology on the one hand, and in mathematical and computational fluid dynamics on the other.

This volume contains a wide-ranging collection of up-to-date, valuable research papers written by some of the most eminent experts in the field. Topics range from fundamental aspects of mathematical fluid mechanics, including topological vortex dynamics and magnetohydrodynamics, integrability issues, Hamiltonian structures and singularity formation, to DNA tangles and knotted DNAs in sedimentation. A substantial introductory chapter on knots and links, covering elements of modern braid theory and knot polynomials, as well as more advanced topics in knot classification, provides an invaluable addition to this material.

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