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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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Fibre bundles, now an integral part of differential geometry, are also of great importance in modern physics--such as in gauge theory. This book, a succinct introduction to the subject by renown mathematician Norman Steenrod, was the first to present the subject systematically. It begins with a general introduction to bundles, including such topics as differentiable manifolds and covering spaces. The author then provides brief surveys of advanced topics, such as homotopy theory and cohomology theory, before using them to study further properties of fibre bundles. The result is a classic and timeless work of great utility that will appeal to serious mathematicians and theoretical physicists alike.

#WWIS:d.d. Prof. L. Bouckaert/ALTO --- 515.1 --- 515.1 Topology --- Topology --- Topology. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Algebraic topology. --- Associated bundle. --- Associative algebra. --- Associative property. --- Atlas (topology). --- Automorphism. --- Axiomatic system. --- Barycentric subdivision. --- Bilinear map. --- Bundle map. --- Classification theorem. --- Coefficient. --- Cohomology ring. --- Cohomology. --- Conjugacy class. --- Connected component (graph theory). --- Connected space. --- Coordinate system. --- Coset. --- Cup product. --- Cyclic group. --- Determinant. --- Differentiable manifold. --- Differential structure. --- Dimension (vector space). --- Direct product. --- Division algebra. --- Equivalence class. --- Equivalence relation. --- Euler number. --- Existence theorem. --- Existential quantification. --- Factorization. --- Fiber bundle. --- Frenet–Serret formulas. --- Gram–Schmidt process. --- Group theory. --- Homeomorphism. --- Homology (mathematics). --- Homomorphism. --- Homotopy group. --- Homotopy. --- Hopf theorem. --- Hurewicz theorem. --- Identity element. --- Inclusion map. --- Inner automorphism. --- Invariant subspace. --- Invertible matrix. --- Jacobian matrix and determinant. --- Klein bottle. --- Lattice of subgroups. --- Lie group. --- Line element. --- Line segment. --- Linear map. --- Linear space (geometry). --- Linear subspace. --- Manifold. --- Mapping cylinder. --- Metric tensor. --- N-sphere. --- Natural topology. --- Octonion. --- Open set. --- Orientability. --- Orthogonal group. --- Orthogonalization. --- Permutation. --- Principal bundle. --- Product topology. --- Quadratic form. --- Quaternion. --- Retract. --- Separable space. --- Set theory. --- Simplicial complex. --- Special case. --- Stiefel manifold. --- Subalgebra. --- Subbase. --- Subgroup. --- Subset. --- Symmetric tensor. --- Tangent bundle. --- Tangent space. --- Tangent vector. --- Tensor field. --- Tensor. --- Theorem. --- Tietze extension theorem. --- Topological group. --- Topological space. --- Transitive relation. --- Transpose. --- Union (set theory). --- Unit sphere. --- Universal bundle. --- Vector field.

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This volume gives a clear introductory account of equivariant cohomology, a central topic in algebraic topology. Assuming readers have taken one semester of manifold theory and a year of algebraic topology, Loring Tu begins with the topological construction of equivariant cohomology, then develops the theory for smooth manifolds with the aid of differential forms. To keep the exposition simple, the equivariant localisation theorem is proven only for a circle action. An appendix gives a proof of the equivariant de Rham theorem, demonstrating that equivariant cohomology can be computed using equivariant differential forms. Examples and calculations illustrate new concepts. Exercises include hints or solutions, making this book suitable for self-study.

Cohomology operations. --- Operations (Algebraic topology) --- Algebraic topology --- Algebraic structure. --- Algebraic topology (object). --- Algebraic topology. --- Algebraic variety. --- Basis (linear algebra). --- Boundary (topology). --- CW complex. --- Cellular approximation theorem. --- Characteristic class. --- Classifying space. --- Coefficient. --- Cohomology ring. --- Cohomology. --- Comparison theorem. --- Complex projective space. --- Continuous function. --- Contractible space. --- Cramer's rule. --- Curvature form. --- De Rham cohomology. --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Differential form. --- Differential geometry. --- Dual basis. --- Equivariant K-theory. --- Equivariant cohomology. --- Equivariant map. --- Euler characteristic. --- Euler class. --- Exponential function. --- Exponential map (Lie theory). --- Exponentiation. --- Exterior algebra. --- Exterior derivative. --- Fiber bundle. --- Fixed point (mathematics). --- Frame bundle. --- Fundamental group. --- Fundamental vector field. --- Group action. --- Group homomorphism. --- Group theory. --- Haar measure. --- Homotopy group. --- Homotopy. --- Hopf fibration. --- Identity element. --- Inclusion map. --- Integral curve. --- Invariant subspace. --- K-theory. --- Lie algebra. --- Lie derivative. --- Lie group action. --- Lie group. --- Lie theory. --- Linear algebra. --- Linear function. --- Local diffeomorphism. --- Manifold. --- Mathematics. --- Matrix group. --- Mayer–Vietoris sequence. --- Module (mathematics). --- Morphism. --- Neighbourhood (mathematics). --- Orthogonal group. --- Oscillatory integral. --- Principal bundle. --- Principal ideal domain. --- Quotient group. --- Quotient space (topology). --- Raoul Bott. --- Representation theory. --- Ring (mathematics). --- Singular homology. --- Spectral sequence. --- Stationary phase approximation. --- Structure constants. --- Sub"ient. --- Subcategory. --- Subgroup. --- Submanifold. --- Submersion (mathematics). --- Symplectic manifold. --- Symplectic vector space. --- Tangent bundle. --- Tangent space. --- Theorem. --- Topological group. --- Topological space. --- Topology. --- Unit sphere. --- Unitary group. --- Universal bundle. --- Vector bundle. --- Vector space. --- Weyl group.

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This work deals with an extension of the classical Littlewood-Paley theory in the context of symmetric diffusion semigroups. In this general setting there are applications to a variety of problems, such as those arising in the study of the expansions coming from second order elliptic operators. A review of background material in Lie groups and martingale theory is included to make the monograph more accessible to the student.

Harmonic analysis. Fourier analysis --- Harmonic analysis --- Semigroups --- 517.986.6 --- Lie groups --- Littlewood-Paley theory --- #WWIS:d.d. Prof. L. Bouckaert/BOUC --- Fourier analysis --- Functions of several real variables --- Group theory --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Harmonic analysis of functions of groups and homogeneous spaces --- Harmonic analysis. --- Littlewood-Paley theory. --- Lie groups. --- Semigroups. --- 517.986.6 Harmonic analysis of functions of groups and homogeneous spaces --- Addition. --- Analytic function. --- Axiom. --- Boundary value problem. --- Central limit theorem. --- Change of variables. --- Circle group. --- Classification theorem. --- Commutative property. --- Compact group. --- Complex analysis. --- Convex set. --- Coset. --- Covering space. --- Derivative. --- Differentiable manifold. --- Differential geometry. --- Differential operator. --- Dimension (vector space). --- Dimension. --- Direct sum. --- E6 (mathematics). --- E7 (mathematics). --- E8 (mathematics). --- Elementary proof. --- Equation. --- Equivalence class. --- Existence theorem. --- Existential quantification. --- Fourier analysis. --- Fourier series. --- Fourier transform. --- Function space. --- General linear group. --- Haar measure. --- Harmonic function. --- Hermite polynomials. --- Hilbert transform. --- Homogeneous space. --- Homomorphism. --- Ideal (ring theory). --- Identity matrix. --- Indecomposability. --- Integral transform. --- Invariant measure. --- Invariant subspace. --- Irreducibility (mathematics). --- Irreducible representation. --- Lebesgue measure. --- Legendre polynomials. --- Lie algebra. --- Lie group. --- Linear combination. --- Linear map. --- Local diffeomorphism. --- Markov process. --- Martingale (probability theory). --- Matrix group. --- Measurable function. --- Measure (mathematics). --- Multiple integral. --- Normal subgroup. --- One-dimensional space. --- Open set. --- Ordinary differential equation. --- Orthogonality. --- Orthonormality. --- Parseval's theorem. --- Partial differential equation. --- Probability space. --- Quadratic form. --- Rank of a group. --- Regular representation. --- Riemannian manifold. --- Riesz transform. --- Schur orthogonality relations. --- Scientific notation. --- Semigroup. --- Sequence. --- Special case. --- Stone–Weierstrass theorem. --- Sturm–Liouville theory. --- Subgroup. --- Subset. --- Summation. --- Tensor algebra. --- Tensor product. --- Theorem. --- Theory. --- Topological group. --- Topological space. --- Torus. --- Trigonometric polynomial. --- Trivial representation. --- Uniform convergence. --- Unitary operator. --- Unitary representation. --- Vector field. --- Vector space. --- Lie, Groupes de --- Analyse harmonique

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The application by Fadeev and Pavlov of the Lax-Phillips scattering theory to the automorphic wave equation led Professors Lax and Phillips to reexamine this development within the framework of their theory. This volume sets forth the results of that work in the form of new or more straightforward treatments of the spectral theory of the Laplace-Beltrami operator over fundamental domains of finite area; the meromorphic character over the whole complex plane of the Eisenstein series; and the Selberg trace formula.CONTENTS: 1. Introduction. 2. An abstract scattering theory. 3. A modified theory for second order equations with an indefinite energy form. 4. The Laplace-Beltrami operator for the modular group. 5. The automorphic wave equation. 6. Incoming and outgoing subspaces for the automorphic wave equations. 7. The scattering matrix for the automorphic wave equation. 8. The general case. 9. The Selberg trace formula.

Harmonic analysis. Fourier analysis --- Automorphic functions --- Scattering (Mathematics) --- Fonctions automorphes --- Dispersion (Mathématiques) --- Automorphic functions. --- Scattering (Mathematics). --- Dispersion (Mathématiques) --- Selberg, Formule de trace de --- Selberg trace formula --- Eisenstein series --- Eisenstein, Séries d' --- Scattering theory (Mathematics) --- Boundary value problems --- Differential equations, Partial --- Scattering operator --- Fuchsian functions --- Functions, Automorphic --- Functions, Fuchsian --- Functions of several complex variables --- Absolute continuity. --- Algebra. --- Analytic continuation. --- Analytic function. --- Annulus (mathematics). --- Asymptotic distribution. --- Automorphic function. --- Bilinear form. --- Boundary (topology). --- Boundary value problem. --- Bounded operator. --- Calculation. --- Cauchy sequence. --- Change of variables. --- Complex plane. --- Conjugacy class. --- Convolution. --- Cusp neighborhood. --- Cyclic group. --- Derivative. --- Differential equation. --- Differential operator. --- Dimension (vector space). --- Dimensional analysis. --- Dirichlet integral. --- Dirichlet series. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Elliptic operator. --- Elliptic partial differential equation. --- Equation. --- Equivalence class. --- Even and odd functions. --- Existential quantification. --- Explicit formula. --- Explicit formulae (L-function). --- Exponential function. --- Fourier transform. --- Function space. --- Functional analysis. --- Functional calculus. --- Fundamental domain. --- Harmonic analysis. --- Hilbert space. --- Hyperbolic partial differential equation. --- Infinitesimal generator (stochastic processes). --- Integral equation. --- Integration by parts. --- Invariant subspace. --- Laplace operator. --- Laplace transform. --- Lebesgue measure. --- Linear differential equation. --- Linear space (geometry). --- Matrix (mathematics). --- Maximum principle. --- Meromorphic function. --- Modular group. --- Neumann boundary condition. --- Norm (mathematics). --- Null vector. --- Number theory. --- Operator theory. --- Orthogonal complement. --- Orthonormal basis. --- Paley–Wiener theorem. --- Partial differential equation. --- Perturbation theory (quantum mechanics). --- Perturbation theory. --- Primitive element (finite field). --- Principal component analysis. --- Projection (linear algebra). --- Quadratic form. --- Removable singularity. --- Representation theorem. --- Resolvent set. --- Riemann hypothesis. --- Riemann surface. --- Riemann zeta function. --- Riesz representation theorem. --- Scatter matrix. --- Scattering theory. --- Schwarz reflection principle. --- Selberg trace formula. --- Self-adjoint. --- Semigroup. --- Sign (mathematics). --- Spectral theory. --- Subgroup. --- Subsequence. --- Summation. --- Support (mathematics). --- Theorem. --- Trace class. --- Trace formula. --- Unitary operator. --- Wave equation. --- Weighted arithmetic mean. --- Winding number. --- Eisenstein, Séries d'. --- Analyse harmonique