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This book provides a comprehensive introduction to modern global variational theory on fibred spaces. It is based on differentiation and integration theory of differential forms on smooth manifolds, and on the concepts of global analysis and geometry such as jet prolongations of manifolds, mappings, and Lie groups. The book will be invaluable for researchers and PhD students in differential geometry, global analysis, differential equations on manifolds, and mathematical physics, and for the readers who wish to undertake further rigorous study in this broad interdisciplinary field.
Operator theory --- Banach spaces. --- Interpolation. --- Function spaces. --- Differential operators. --- Banach spaces --- Interpolation --- Function spaces --- Differential operators --- 517.982 --- 517.982 Linear spaces with topology and order or other structures --- Linear spaces with topology and order or other structures --- Approximation theory --- Numerical analysis --- Spaces, Function --- Functional analysis --- Operators, Differential --- Differential equations --- Functions of complex variables --- Generalized spaces --- Topology
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Operator theory --- Opérateurs différentiels. --- Differential operators. --- Analyse fonctionnelle --- Functional analysis --- Rings (Algebra). --- Differential operators --- RINGS (Algebra) --- Rings (Algebra) --- 517.982.4 --- 517.982.4 Theory of generalized functions (distributions) --- Theory of generalized functions (distributions) --- Algebraic rings --- Ring theory --- Algebraic fields --- Operators, Differential --- Differential equations --- Functional analysis. --- Opérateurs différentiels --- Equations aux derivees partielles sur une variete
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Operator theory --- Differential operators --- Selfadjoint operators --- Spectral theory (Mathematics) --- 51 --- Boundary value problems --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Operators, Selfadjoint --- Self-adjoint operators --- Linear operators --- Operators, Differential --- Differential equations --- Boundary conditions (Differential equations) --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Mathematics --- 51 Mathematics --- Équations différentielles --- Théorie spectrale (mathématiques) --- Opérateurs linéaires --- Opérateurs auto-adjoints --- Differential equations. --- Linear operators. --- Selfadjoint operators. --- Équations différentielles. --- Opérateurs auto-adjoints --- Opérateurs linéaires --- Théorie spectrale (mathématiques) --- Problèmes aux limites --- Sturm liouville theory
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One of the basic ideas in differential geometry is that the study of analytic properties of certain differential operators acting on sections of vector bundles yields geometric and topological properties of the underlying base manifold. Symplectic spinor fields are sections in an L^2-Hilbert space bundle over a symplectic manifold and symplectic Dirac operators, acting on symplectic spinor fields, are associated to the symplectic manifold in a very natural way. Hence they may be expected to give interesting applications in symplectic geometry and symplectic topology. These symplectic Dirac operators are called Dirac operators, since they are defined in an analogous way as the classical Riemannian Dirac operator known from Riemannian spin geometry. They are called symplectic because they are constructed by use of the symplectic setting of the underlying symplectic manifold. This volume is the first one that gives a systematic and self-contained introduction to the theory of symplectic Dirac operators and reflects the current state of the subject. At the same time, it is intended to establish the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology, which have become important fields and very active areas of mathematical research.
Symplectic geometry. --- Symplectic and contact topology. --- Symplectic groups. --- Dirac equation. --- Géométrie symplectique --- Topologie symplectique et de contact --- Groupes symplectiques --- Dirac, Equation de --- Symplectic geometry --- Symplectic and contact topology --- Symplectic groups --- Dirac equation --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Algebra --- Clifford algebras. --- Differential operators. --- Operators, Differential --- Geometric algebras --- Mathematics. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Differential geometry. --- Differential Geometry. --- Global Analysis and Analysis on Manifolds. --- Differential geometry --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Math --- Science --- Differential equations --- Operator theory --- Differential equations, Partial --- Quantum field theory --- Wave equation --- Algebras, Linear --- Global differential geometry. --- Global analysis. --- Global analysis (Mathematics) --- Groups, Symplectic --- Linear algebraic groups --- Topology, Symplectic and contact
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