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The classical uniformization theorem for Riemann surfaces and its recent extensions can be viewed as introducing special pseudogroup structures, affine or projective structures, on Riemann surfaces. In fact, the additional structures involved can be considered as local forms of the uniformizations of Riemann surfaces. In this study, Robert Gunning discusses the corresponding pseudogroup structures on higher-dimensional complex manifolds, modeled on the theory as developed for Riemann surfaces.Originally published in 1978.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.

Analytical spaces --- Differential geometry. Global analysis --- Complex manifolds --- Connections (Mathematics) --- Pseudogroups --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Global analysis (Mathematics) --- Lie groups --- Geometry, Differential --- Analytic spaces --- Manifolds (Mathematics) --- Adjunction formula. --- Affine connection. --- Affine transformation. --- Algebraic surface. --- Algebraic torus. --- Algebraic variety. --- Analytic continuation. --- Analytic function. --- Automorphic function. --- Automorphism. --- Bilinear form. --- Canonical bundle. --- Characterization (mathematics). --- Cohomology. --- Compact Riemann surface. --- Complex Lie group. --- Complex analysis. --- Complex dimension. --- Complex manifold. --- Complex multiplication. --- Complex number. --- Complex plane. --- Complex torus. --- Complex vector bundle. --- Contraction mapping. --- Covariant derivative. --- Differentiable function. --- Differentiable manifold. --- Differential equation. --- Differential form. --- Differential geometry. --- Differential operator. --- Dimension (vector space). --- Dimension. --- Elliptic operator. --- Elliptic surface. --- Enriques surface. --- Equation. --- Existential quantification. --- Explicit formula. --- Explicit formulae (L-function). --- Exterior derivative. --- Fiber bundle. --- General linear group. --- Geometric genus. --- Group homomorphism. --- Hausdorff space. --- Holomorphic function. --- Homomorphism. --- Identity matrix. --- Invariant subspace. --- Invertible matrix. --- Irreducible representation. --- Jacobian matrix and determinant. --- K3 surface. --- Kähler manifold. --- Lie algebra representation. --- Lie algebra. --- Line bundle. --- Linear equation. --- Linear map. --- Linear space (geometry). --- Linear subspace. --- Manifold. --- Mathematical analysis. --- Mathematical induction. --- Ordinary differential equation. --- Partial differential equation. --- Permutation. --- Polynomial. --- Principal bundle. --- Projection (linear algebra). --- Projective connection. --- Projective line. --- Pseudogroup. --- Quadratic transformation. --- Quotient space (topology). --- Representation theory. --- Riemann surface. --- Riemann–Roch theorem. --- Schwarzian derivative. --- Sheaf (mathematics). --- Special case. --- Subalgebra. --- Subgroup. --- Submanifold. --- Symmetric tensor. --- Symmetrization. --- Tangent bundle. --- Tangent space. --- Tensor field. --- Tensor product. --- Tensor. --- Theorem. --- Topological manifold. --- Uniformization theorem. --- Uniformization. --- Unit (ring theory). --- Vector bundle. --- Vector space. --- Fonctions de plusieurs variables complexes --- Variétés complexes

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This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrödinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. The physical motivation of these models extends back to the works of Rudolph Peierls and Douglas R. Hofstadter, and the models themselves have been a focus of mathematical research for two decades. Jean Bourgain here sets forth the results and techniques that have been discovered in the last few years. He puts special emphasis on so-called "non-perturbative" methods and the important role of subharmonic function theory and semi-algebraic set methods. He describes various applications to the theory of differential equations and dynamical systems, in particular to the quantum kicked rotor and KAM theory for nonlinear Hamiltonian evolution equations. Intended primarily for graduate students and researchers in the general area of dynamical systems and mathematical physics, the book provides a coherent account of a large body of work that is presently scattered in the literature. It does so in a refreshingly contained manner that seeks to convey the present technological "state of the art."

Schrödinger operator. --- Green's functions. --- Hamiltonian systems. --- Evolution equations. --- Evolutionary equations --- Equations, Evolution --- Equations of evolution --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Functions, Green's --- Functions, Induction --- Functions, Source --- Green functions --- Induction functions --- Source functions --- Operator, Schrödinger --- Differential equations --- Differentiable dynamical systems --- Potential theory (Mathematics) --- Differential operators --- Quantum theory --- Schrödinger equation --- Almost Mathieu operator. --- Analytic function. --- Anderson localization. --- Betti number. --- Cartan's theorem. --- Chaos theory. --- Density of states. --- Dimension (vector space). --- Diophantine equation. --- Dynamical system. --- Equation. --- Existential quantification. --- Fundamental matrix (linear differential equation). --- Green's function. --- Hamiltonian system. --- Hermitian adjoint. --- Infimum and supremum. --- Iterative method. --- Jacobi operator. --- Linear equation. --- Linear map. --- Linearization. --- Monodromy matrix. --- Non-perturbative. --- Nonlinear system. --- Normal mode. --- Parameter space. --- Parameter. --- Parametrization. --- Partial differential equation. --- Periodic boundary conditions. --- Phase space. --- Phase transition. --- Polynomial. --- Renormalization. --- Self-adjoint. --- Semialgebraic set. --- Special case. --- Statistical significance. --- Subharmonic function. --- Summation. --- Theorem. --- Theory. --- Transfer matrix. --- Transversality (mathematics). --- Trigonometric functions. --- Trigonometric polynomial. --- Uniformization theorem.

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Intended for researchers in Riemann surfaces, this volume summarizes a significant portion of the work done in the field during the years 1966 to 1971.

Riemann surfaces --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Surfaces, Riemann --- Functions --- Congresses --- Differential geometry. Global analysis --- RIEMANN SURFACES --- congresses --- Congresses. --- MATHEMATICS / Calculus. --- Affine space. --- Algebraic function field. --- Algebraic structure. --- Analytic continuation. --- Analytic function. --- Analytic set. --- Automorphic form. --- Automorphic function. --- Automorphism. --- Beltrami equation. --- Bernhard Riemann. --- Boundary (topology). --- Canonical basis. --- Cartesian product. --- Clifford's theorem. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Complex multiplication. --- Conformal geometry. --- Conformal map. --- Coset. --- Degeneracy (mathematics). --- Diagram (category theory). --- Differential geometry of surfaces. --- Dimension (vector space). --- Dirichlet boundary condition. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Euclidean space. --- Existential quantification. --- Explicit formulae (L-function). --- Exterior (topology). --- Finsler manifold. --- Fourier series. --- Fuchsian group. --- Function (mathematics). --- Generating set of a group. --- Group (mathematics). --- Hilbert space. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Homotopy. --- Hyperbolic geometry. --- Hyperbolic group. --- Identity matrix. --- Infimum and supremum. --- Inner automorphism. --- Intersection (set theory). --- Intersection number (graph theory). --- Isometry. --- Isomorphism class. --- Isomorphism theorem. --- Kleinian group. --- Limit point. --- Limit set. --- Linear map. --- Lorentz group. --- Mapping class group. --- Mathematical induction. --- Mathematics. --- Matrix (mathematics). --- Matrix multiplication. --- Measure (mathematics). --- Meromorphic function. --- Metric space. --- Modular group. --- Möbius transformation. --- Number theory. --- Osgood curve. --- Parity (mathematics). --- Partial isometry. --- Poisson summation formula. --- Pole (complex analysis). --- Projective space. --- Quadratic differential. --- Quadratic form. --- Quasiconformal mapping. --- Quotient space (linear algebra). --- Quotient space (topology). --- Riemann mapping theorem. --- Riemann sphere. --- Riemann surface. --- Riemann zeta function. --- Scalar multiplication. --- Scientific notation. --- Selberg trace formula. --- Series expansion. --- Sign (mathematics). --- Square-integrable function. --- Subgroup. --- Teichmüller space. --- Theorem. --- Topological manifold. --- Topological space. --- Uniformization. --- Unit disk. --- Variable (mathematics). --- Riemann, Surfaces de --- RIEMANN SURFACES - congresses --- Fonctions d'une variable complexe --- Surfaces de riemann

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Based on seven lecture series given by leading experts at a summer school at Peking University, in Beijing, in 1984. this book surveys recent developments in the areas of harmonic analysis most closely related to the theory of singular integrals, real-variable methods, and applications to several complex variables and partial differential equations. The different lecture series are closely interrelated; each contains a substantial amount of background material, as well as new results not previously published. The contributors to the volume are R. R. Coifman and Yves Meyer, Robert Fcfferman,Carlos K. Kenig, Steven G. Krantz, Alexander Nagel, E. M. Stein, and Stephen Wainger.

Harmonic analysis. --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Analytic function. --- Asymptotic formula. --- Bergman metric. --- Bernhard Riemann. --- Bessel function. --- Biholomorphism. --- Boundary value problem. --- Bounded mean oscillation. --- Bounded operator. --- Boundedness. --- Cauchy's integral formula. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Coefficient. --- Commutator. --- Complexification (Lie group). --- Continuous function. --- Convolution. --- Degeneracy (mathematics). --- Differential equation. --- Differential operator. --- Dirac delta function. --- Dirichlet problem. --- Equation. --- Estimation. --- Existence theorem. --- Existential quantification. --- Explicit formula. --- Explicit formulae (L-function). --- Fatou's theorem. --- Fourier analysis. --- Fourier integral operator. --- Fourier transform. --- Fredholm theory. --- Fubini's theorem. --- Function (mathematics). --- Functional calculus. --- Fundamental solution. --- Gaussian curvature. --- Hardy space. --- Harmonic function. --- Harmonic measure. --- Heisenberg group. --- Hilbert space. --- Hilbert transform. --- Hodge theory. --- Holomorphic function. --- Hyperbolic partial differential equation. --- Hölder's inequality. --- Infimum and supremum. --- Integration by parts. --- Interpolation theorem. --- Intersection (set theory). --- Invertible matrix. --- Isometry group. --- Laplace operator. --- Laplace's equation. --- Lebesgue measure. --- Linear map. --- Lipschitz continuity. --- Lipschitz domain. --- Lp space. --- Mathematical induction. --- Mathematical physics. --- Maximal function. --- Maximum principle. --- Measure (mathematics). --- Newtonian potential. --- Non-Euclidean geometry. --- Number theory. --- Operator theory. --- Oscillatory integral. --- Parameter. --- Partial derivative. --- Partial differential equation. --- Polynomial. --- Power series. --- Product metric. --- Radon–Nikodym theorem. --- Riemannian manifold. --- Riesz representation theorem. --- Scientific notation. --- Several complex variables. --- Sign (mathematics). --- Simultaneous equations. --- Singular function. --- Singular integral. --- Sobolev space. --- Square (algebra). --- Statistical hypothesis testing. --- Stokes' theorem. --- Support (mathematics). --- Tangent space. --- Tensor product. --- Theorem. --- Trigonometric series. --- Uniformization theorem. --- Variable (mathematics). --- Vector field.

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Modular Forms and Special Cycles on Shimura Curves is a thorough study of the generating functions constructed from special cycles, both divisors and zero-cycles, on the arithmetic surface "M" attached to a Shimura curve "M" over the field of rational numbers. These generating functions are shown to be the q-expansions of modular forms and Siegel modular forms of genus two respectively, valued in the Gillet-Soulé arithmetic Chow groups of "M". The two types of generating functions are related via an arithmetic inner product formula. In addition, an analogue of the classical Siegel-Weil formula identifies the generating function for zero-cycles as the central derivative of a Siegel Eisenstein series. As an application, an arithmetic analogue of the Shimura-Waldspurger correspondence is constructed, carrying holomorphic cusp forms of weight 3/2 to classes in the Mordell-Weil group of "M". In certain cases, the nonvanishing of this correspondence is related to the central derivative of the standard L-function for a modular form of weight 2. These results depend on a novel mixture of modular forms and arithmetic geometry and should provide a paradigm for further investigations. The proofs involve a wide range of techniques, including arithmetic intersection theory, the arithmetic adjunction formula, representation densities of quadratic forms, deformation theory of p-divisible groups, p-adic uniformization, the Weil representation, the local and global theta correspondence, and the doubling integral representation of L-functions.

Arithmetical algebraic geometry. --- Shimura varieties. --- Varieties, Shimura --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Arithmetical algebraic geometry --- Number theory --- Abelian group. --- Addition. --- Adjunction formula. --- Algebraic number theory. --- Arakelov theory. --- Arithmetic. --- Automorphism. --- Bijection. --- Borel subgroup. --- Calculation. --- Chow group. --- Coefficient. --- Cohomology. --- Combinatorics. --- Compact Riemann surface. --- Complex multiplication. --- Complex number. --- Cup product. --- Deformation theory. --- Derivative. --- Dimension. --- Disjoint union. --- Divisor. --- Dual pair. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Elliptic curve. --- Endomorphism. --- Equation. --- Explicit formulae (L-function). --- Fields Institute. --- Formal group. --- Fourier series. --- Fundamental matrix (linear differential equation). --- Galois group. --- Generating function. --- Green's function. --- Group action. --- Induced representation. --- Intersection (set theory). --- Intersection number. --- Irreducible component. --- Isomorphism class. --- L-function. --- Laurent series. --- Level structure. --- Line bundle. --- Local ring. --- Mathematical sciences. --- Mathematics. --- Metaplectic group. --- Modular curve. --- Modular form. --- Modularity (networks). --- Moduli space. --- Multiple integral. --- Number theory. --- Numerical integration. --- Orbifold. --- Orthogonal complement. --- P-adic number. --- Pairing. --- Prime factor. --- Prime number. --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Quadratic form. --- Quadratic residue. --- Quantity. --- Quaternion algebra. --- Quaternion. --- Quotient stack. --- Rational number. --- Real number. --- Residue field. --- Riemann zeta function. --- Ring of integers. --- SL2(R). --- Scientific notation. --- Shimura variety. --- Siegel Eisenstein series. --- Siegel modular form. --- Special case. --- Standard L-function. --- Subgroup. --- Subset. --- Summation. --- Tensor product. --- Test vector. --- Theorem. --- Three-dimensional space (mathematics). --- Topology. --- Trace (linear algebra). --- Triangular matrix. --- Two-dimensional space. --- Uniformization. --- Valuative criterion. --- Whittaker function.