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Differential geometry. Global analysis --- Geometry, Differential --- Géométrie différentielle --- 514.75 --- Differential geometry --- Differential geometry in spaces with fundamental groups --- 514.75 Differential geometry in spaces with fundamental groups --- Géométrie différentielle --- Geometry, Differential. --- Hypersurfaces in differential geometry --- Integral manifolds --- Lie groups --- Moving frame method --- Tangent bundle --- Tensors --- Topology(Algebraic-) --- Gauss Bonnet theorem --- Gauss theory of surfaces --- Ricci calculus --- Riemann curvature tensor --- Riemann metric

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The aim of this work is to provide a proof of the nonlinear gravitational stability of the Minkowski space-time. More precisely, the book offers a constructive proof of global, smooth solutions to the Einstein Vacuum Equations, which look, in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singularities. The work contains a detailed description of the sense in which these solutions are close to the Minkowski space-time, in all directions. It thus provides the mathematical framework in which we can give a rigorous derivation of the laws of gravitation proposed by Bondi. Moreover, it establishes other important conclusions concerning the nonlinear character of gravitational radiation. The authors obtain their solutions as dynamic developments of all initial data sets, which are close, in a precise manner, to the flat initial data set corresponding to the Minkowski space-time. They thus establish the global dynamic stability of the latter.Originally published in 1994.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Space and time --- Generalized spaces --- Nonlinear theories --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Nonlinear problems --- Nonlinearity (Mathematics) --- Calculus --- Mathematical analysis --- Mathematical physics --- Geometry of paths --- Minkowski space --- Spaces, Generalized --- Weyl space --- Calculus of tensors --- Geometry, Differential --- Geometry, Non-Euclidean --- Hyperspace --- Relativity (Physics) --- Space of more than three dimensions --- Space-time --- Space-time continuum --- Space-times --- Spacetime --- Time and space --- Fourth dimension --- Infinite --- Metaphysics --- Philosophy --- Space sciences --- Time --- Beginning --- Mathematics --- Angular momentum operator. --- Asymptotic analysis. --- Asymptotic expansion. --- Big O notation. --- Boundary value problem. --- Cauchy–Riemann equations. --- Coarea formula. --- Coefficient. --- Compactification (mathematics). --- Comparison theorem. --- Corollary. --- Covariant derivative. --- Curvature tensor. --- Curvature. --- Cut locus (Riemannian manifold). --- Degeneracy (mathematics). --- Degrees of freedom (statistics). --- Derivative. --- Diffeomorphism. --- Differentiable function. --- Eigenvalues and eigenvectors. --- Eikonal equation. --- Einstein field equations. --- Equation. --- Error term. --- Estimation. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- Exponential map (Lie theory). --- Exponential map (Riemannian geometry). --- Exterior (topology). --- Foliation. --- Fréchet derivative. --- Geodesic curvature. --- Geodesic. --- Geodesics in general relativity. --- Geometry. --- Hodge dual. --- Homotopy. --- Hyperbolic partial differential equation. --- Hypersurface. --- Hölder's inequality. --- Identity (mathematics). --- Infinitesimal generator (stochastic processes). --- Integral curve. --- Intersection (set theory). --- Isoperimetric inequality. --- Laplace's equation. --- Lie algebra. --- Lie derivative. --- Linear equation. --- Linear map. --- Logarithm. --- Lorentz group. --- Lp space. --- Mass formula. --- Mean curvature. --- Metric tensor. --- Minkowski space. --- Nonlinear system. --- Normal (geometry). --- Null hypersurface. --- Orthonormal basis. --- Partial derivative. --- Poisson's equation. --- Projection (linear algebra). --- Quantity. --- Radial function. --- Ricci curvature. --- Riemann curvature tensor. --- Riemann surface. --- Riemannian geometry. --- Riemannian manifold. --- Sard's theorem. --- Scalar (physics). --- Scalar curvature. --- Scale invariance. --- Schwarzschild metric. --- Second derivative. --- Second fundamental form. --- Sobolev inequality. --- Sobolev space. --- Stokes formula. --- Stokes' theorem. --- Stress–energy tensor. --- Symmetric tensor. --- Symmetrization. --- Tangent space. --- Tensor product. --- Theorem. --- Trace (linear algebra). --- Transversal (geometry). --- Triangle inequality. --- Uniformization theorem. --- Unit sphere. --- Vector field. --- Volume element. --- Wave equation. --- Weyl tensor.

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This book presents the analytic foundations to the theory of the hypoelliptic Laplacian. The hypoelliptic Laplacian, a second-order operator acting on the cotangent bundle of a compact manifold, is supposed to interpolate between the classical Laplacian and the geodesic flow. Jean-Michel Bismut and Gilles Lebeau establish the basic functional analytic properties of this operator, which is also studied from the perspective of local index theory and analytic torsion. The book shows that the hypoelliptic Laplacian provides a geometric version of the Fokker-Planck equations. The authors give the proper functional analytic setting in order to study this operator and develop a pseudodifferential calculus, which provides estimates on the hypoelliptic Laplacian's resolvent. When the deformation parameter tends to zero, the hypoelliptic Laplacian converges to the standard Hodge Laplacian of the base by a collapsing argument in which the fibers of the cotangent bundle collapse to a point. For the local index theory, small time asymptotics for the supertrace of the associated heat kernel are obtained. The Ray-Singer analytic torsion of the hypoelliptic Laplacian as well as the associated Ray-Singer metrics on the determinant of the cohomology are studied in an equivariant setting, resulting in a key comparison formula between the elliptic and hypoelliptic analytic torsions.

Differential equations, Hypoelliptic. --- Laplacian operator. --- Metric spaces. --- Spaces, Metric --- Operator, Laplacian --- Hypoelliptic differential equations --- Generalized spaces --- Set theory --- Topology --- Differential equations, Partial --- Alexander Grothendieck. --- Analytic function. --- Asymptote. --- Asymptotic expansion. --- Berezin integral. --- Bijection. --- Brownian dynamics. --- Brownian motion. --- Chaos theory. --- Chern class. --- Classical Wiener space. --- Clifford algebra. --- Cohomology. --- Combination. --- Commutator. --- Computation. --- Connection form. --- Coordinate system. --- Cotangent bundle. --- Covariance matrix. --- Curvature tensor. --- Curvature. --- De Rham cohomology. --- Derivative. --- Determinant. --- Differentiable manifold. --- Differential operator. --- Dirac operator. --- Direct proof. --- Eigenform. --- Eigenvalues and eigenvectors. --- Ellipse. --- Embedding. --- Equation. --- Estimation. --- Euclidean space. --- Explicit formula. --- Explicit formulae (L-function). --- Feynman–Kac formula. --- Fiber bundle. --- Fokker–Planck equation. --- Formal power series. --- Fourier series. --- Fourier transform. --- Fredholm determinant. --- Function space. --- Girsanov theorem. --- Ground state. --- Heat kernel. --- Hilbert space. --- Hodge theory. --- Holomorphic function. --- Holomorphic vector bundle. --- Hypoelliptic operator. --- Integration by parts. --- Invertible matrix. --- Logarithm. --- Malliavin calculus. --- Martingale (probability theory). --- Matrix calculus. --- Mellin transform. --- Morse theory. --- Notation. --- Parameter. --- Parametrix. --- Parity (mathematics). --- Polynomial. --- Principal bundle. --- Probabilistic method. --- Projection (linear algebra). --- Rectangle. --- Resolvent set. --- Ricci curvature. --- Riemann–Roch theorem. --- Scientific notation. --- Self-adjoint operator. --- Self-adjoint. --- Sign convention. --- Smoothness. --- Sobolev space. --- Spectral theory. --- Square root. --- Stochastic calculus. --- Stochastic process. --- Summation. --- Supertrace. --- Symmetric space. --- Tangent space. --- Taylor series. --- Theorem. --- Theory. --- Torus. --- Trace class. --- Translational symmetry. --- Transversality (mathematics). --- Uniform convergence. --- Variable (mathematics). --- Vector bundle. --- Vector space. --- Wave equation.

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Emergent quantum mechanics explores the possibility of an ontology for quantum mechanics. The resurgence of interest in ""deeper-level"" theories for quantum phenomena challenges the standard, textbook interpretation. The book presents expert views that critically evaluate the significance—for 21st century physics—of ontological quantum mechanics, an approach that David Bohm helped pioneer. The possibility of a deterministic quantum theory was first introduced with the original de Broglie-Bohm theory, which has also been developed as Bohmian mechanics. The wide range of perspectives that were contributed to this book on the occasion of David Bohm’s centennial celebration provide ample evidence for the physical consistency of ontological quantum mechanics. The book addresses deeper-level questions such as the following: Is reality intrinsically random or fundamentally interconnected? Is the universe local or nonlocal? Might a radically new conception of reality include a form of quantum causality or quantum ontology? What is the role of the experimenter agent? As the book demonstrates, the advancement of ‘quantum ontology’—as a scientific concept—marks a clear break with classical reality. The search for quantum reality entails unconventional causal structures and non-classical ontology, which can be fully consistent with the known record of quantum observations in the laboratory.

non-locality --- ultraviolet divergence --- constraints --- Kilmister equation --- bohmian mechanics --- epistemic agent --- Bohmian mechanics --- relational space --- Feynman paths --- Langevin equation --- quantum causality --- emergent quantum gravity --- quantum ontology --- interpretations --- emergent quantum state --- undecidable dynamics --- molecule interference --- emergent quantum mechanics --- no-hidden-variables theorems --- mind–body problem --- physical ontology --- quantum foundations --- matter-wave optics --- conscious agent --- diffusion constant --- Bell theorem --- Burgers equation --- objective non-signaling constraint --- self-referential dynamics --- Bell inequality --- interpretation --- photochemistry --- Born rule statistics --- sub-quantum dynamics --- dynamical chaos --- weak measurement --- p-adic metric --- Levi-Civita connection --- David Bohm --- H-theorem --- the causal arrow of time --- strong coupling --- vortical dynamics --- fundamental irreversibility --- magnetic deflectometry --- quantum thermodynamics --- de Broglie–Bohm interpretation of quantum mechanics --- wavefunction nodes --- stochastic quantum dynamics --- entropic gravity --- metrology --- Schrödinger equation --- gauge freedom --- Monte Carlo simulations --- micro-constituents --- nonequilibrium thermodynamics --- Bell’s theorem --- emergent space-time --- spin --- quantum field theory --- time-symmetry --- Gaussian-like solutions --- Hamiltonian --- number theory --- fractional velocity --- ergodicity --- fractal geometry --- atomic metastable states --- operator thermodynamic functions --- Canonical Presentation --- Retrocausation --- interpretations of quantum mechanics --- Bohm theory --- quantum mechanics --- zero-point field --- conspiracy --- pilot wave --- quantum holism --- toy-models --- curvature tensor --- Aharonov–Bohm effect --- computational irreducibility --- Stochastic Electrodynamics --- diffraction --- retrocausality --- resonances in quantum systems --- stochastic differential equations --- Bianchi identity --- past of the photon --- commutator --- relational interpretation of quantum mechanics --- free will --- nomology --- trajectories --- primitive ontology --- Mach–Zehnder interferometer --- weak values --- singular limit --- interior-boundary condition --- Poincaré recurrence --- quantum inaccessibility --- symplectic camel --- surrealistic trajectories --- observables --- Stern-Gerlach --- decoherence --- quantum non-equilibrium --- generalized Lagrangian paths --- superdeterminism --- black hole thermodynamics --- nonlocality --- measurement problem --- entropy and time evolution --- bouncing oil droplets --- spontaneous state reduction --- quantum theory --- many interacting worlds --- complex entropy. --- Turing incomputability --- iterant --- space-time fluctuations --- quantum potential --- ontological quantum mechanics --- photon trajectory --- Dove prism --- the Friedrichs model --- contextuality --- discrete calculus --- transition probability amplitude --- gravity --- pilot-wave theory --- matter-waves --- de Broglie-Bohm theory --- covariant quantum gravity --- atom-surface scattering --- de Broglie–Bohm theory --- non-locality --- ultraviolet divergence --- constraints --- Kilmister equation --- bohmian mechanics --- epistemic agent --- Bohmian mechanics --- relational space --- Feynman paths --- Langevin equation --- quantum causality --- emergent quantum gravity --- quantum ontology --- interpretations --- emergent quantum state --- undecidable dynamics --- molecule interference --- emergent quantum mechanics --- no-hidden-variables theorems --- mind–body problem --- physical ontology --- quantum foundations --- matter-wave optics --- conscious agent --- diffusion constant --- Bell theorem --- Burgers equation --- objective non-signaling constraint --- self-referential dynamics --- Bell inequality --- interpretation --- photochemistry --- Born rule statistics --- sub-quantum dynamics --- dynamical chaos --- weak measurement --- p-adic metric --- Levi-Civita connection --- David Bohm --- H-theorem --- the causal arrow of time --- strong coupling --- vortical dynamics --- fundamental irreversibility --- magnetic deflectometry --- quantum thermodynamics --- de Broglie–Bohm interpretation of quantum mechanics --- wavefunction nodes --- stochastic quantum dynamics --- entropic gravity --- metrology --- Schrödinger equation --- gauge freedom --- Monte Carlo simulations --- micro-constituents --- nonequilibrium thermodynamics --- Bell’s theorem --- emergent space-time --- spin --- quantum field theory --- time-symmetry --- Gaussian-like solutions --- Hamiltonian --- number theory --- fractional velocity --- ergodicity --- fractal geometry --- atomic metastable states --- operator thermodynamic functions --- Canonical Presentation --- Retrocausation --- interpretations of quantum mechanics --- Bohm theory --- quantum mechanics --- zero-point field --- conspiracy --- pilot wave --- quantum holism --- toy-models --- curvature tensor --- Aharonov–Bohm effect --- computational irreducibility --- Stochastic Electrodynamics --- diffraction --- retrocausality --- resonances in quantum systems --- stochastic differential equations --- Bianchi identity --- past of the photon --- commutator --- relational interpretation of quantum mechanics --- free will --- nomology --- trajectories --- primitive ontology --- Mach–Zehnder interferometer --- weak values --- singular limit --- interior-boundary condition --- Poincaré recurrence --- quantum inaccessibility --- symplectic camel --- surrealistic trajectories --- observables --- Stern-Gerlach --- decoherence --- quantum non-equilibrium --- generalized Lagrangian paths --- superdeterminism --- black hole thermodynamics --- nonlocality --- measurement problem --- entropy and time evolution --- bouncing oil droplets --- spontaneous state reduction --- quantum theory --- many interacting worlds --- complex entropy. --- Turing incomputability --- iterant --- space-time fluctuations --- quantum potential --- ontological quantum mechanics --- photon trajectory --- Dove prism --- the Friedrichs model --- contextuality --- discrete calculus --- transition probability amplitude --- gravity --- pilot-wave theory --- matter-waves --- de Broglie-Bohm theory --- covariant quantum gravity --- atom-surface scattering --- de Broglie–Bohm theory

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This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.

Algebres de Clifford --- Clifford [Algebra's van ] --- Clifford algebras --- Fysica [Mathematische ] --- Fysica [Wiskundige ] --- Mathematische fysica --- Physics -- Mathematics --- Physics [Mathematical ] --- Physique -- Mathématiques --- Physique -- Méthodes mathématiques --- Wiskundige fysica --- Clifford, Algèbres de --- Spin, Nuclear --- Geometric algebras --- Clifford algebras. --- Spin geometry. --- Clifford, Algèbres de --- Spin geometry --- 514.76 --- Algebras, Linear --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Geometry of differentiable manifolds and of their submanifolds --- Global differential geometry --- Geometry --- Mathematical physics --- Topology --- Nuclear spin --- -Mathematics --- Géométrie --- Physique mathématique --- Spin nucléaire --- Topologie --- Mathematics --- Mathématiques --- Algebraic theory. --- Atiyah–Singer index theorem. --- Automorphism. --- Betti number. --- Binary icosahedral group. --- Binary octahedral group. --- Bundle metric. --- C*-algebra. --- Calabi conjecture. --- Calabi–Yau manifold. --- Cartesian product. --- Classification theorem. --- Clifford algebra. --- Cobordism. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Complete metric space. --- Complex manifold. --- Complex vector bundle. --- Complexification (Lie group). --- Covering space. --- Diffeomorphism. --- Differential topology. --- Dimension (vector space). --- Dimension. --- Dirac operator. --- Disk (mathematics). --- Dolbeault cohomology. --- Einstein field equations. --- Elliptic operator. --- Equivariant K-theory. --- Exterior algebra. --- Fiber bundle. --- Fixed-point theorem. --- Fourier inversion theorem. --- Fundamental group. --- Gauge theory. --- Geometry. --- Hilbert scheme. --- Holonomy. --- Homotopy sphere. --- Homotopy. --- Hyperbolic manifold. --- Induced homomorphism. --- Intersection form (4-manifold). --- Isomorphism class. --- J-invariant. --- K-theory. --- Kähler manifold. --- Laplace operator. --- Lie algebra. --- Lorentz covariance. --- Lorentz group. --- Manifold. --- Mathematical induction. --- Metric connection. --- Minkowski space. --- Module (mathematics). --- N-sphere. --- Operator (physics). --- Orthonormal basis. --- Principal bundle. --- Projective space. --- Pseudo-Riemannian manifold. --- Pseudo-differential operator. --- Quadratic form. --- Quaternion. --- Quaternionic projective space. --- Ricci curvature. --- Riemann curvature tensor. --- Riemannian geometry. --- Riemannian manifold. --- Ring homomorphism. --- Scalar curvature. --- Scalar multiplication. --- Sign (mathematics). --- Space form. --- Sphere theorem. --- Spin representation. --- Spin structure. --- Spinor bundle. --- Spinor field. --- Spinor. --- Subgroup. --- Support (mathematics). --- Symplectic geometry. --- Tangent bundle. --- Tangent space. --- Tensor calculus. --- Tensor product. --- Theorem. --- Topology. --- Unit disk. --- Unit sphere. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume form. --- Nuclear spin - - Mathematics --- -Clifford algebras.