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The aim of this book is to study harmonic maps, minimal and parallel mean curvature immersions in the presence of symmetry. In several instances, the latter permits reduction of the original elliptic variational problem to the qualitative study of certain ordinary differential equations: the authors' primary objective is to provide representative examples to illustrate these reduction methods and their associated analysis with geometric and topological applications. The material covered by the book displays a solid interplay involving geometry, analysis and topology: in particular, it includes a basic presentation of 1-cohomogeneous equivariant differential geometry and of the theory of harmonic maps between spheres.

Cartes harmoniques --- Harmonic maps --- Harmonische kaarten --- Immersies (Wiskunde) --- Immersions (Mathematics) --- Immersions (Mathématiques) --- Harmonic maps. --- Differential equations, Elliptic --- Applications harmoniques --- Immersions (Mathematiques) --- Équations différentielles elliptiques --- Numerical solutions. --- Solutions numériques --- Équations différentielles elliptiques --- Solutions numériques --- Differential equations [Elliptic] --- Numerical solutions --- Embeddings (Mathematics) --- Manifolds (Mathematics) --- Mappings (Mathematics) --- Maps, Harmonic --- Arc length. --- Catenary. --- Clifford algebra. --- Codimension. --- Coefficient. --- Compact space. --- Complex projective space. --- Connected sum. --- Constant curvature. --- Corollary. --- Covariant derivative. --- Curvature. --- Cylinder (geometry). --- Degeneracy (mathematics). --- Diagram (category theory). --- Differential equation. --- Differential geometry. --- Elliptic partial differential equation. --- Embedding. --- Energy functional. --- Equation. --- Existence theorem. --- Existential quantification. --- Fiber bundle. --- Gauss map. --- Geometry and topology. --- Geometry. --- Gravitational field. --- Harmonic map. --- Hyperbola. --- Hyperplane. --- Hypersphere. --- Hypersurface. --- Integer. --- Iterative method. --- Levi-Civita connection. --- Lie group. --- Mathematics. --- Maximum principle. --- Mean curvature. --- Normal (geometry). --- Numerical analysis. --- Open set. --- Ordinary differential equation. --- Parabola. --- Quadratic form. --- Sign (mathematics). --- Special case. --- Stiefel manifold. --- Submanifold. --- Suggestion. --- Surface of revolution. --- Symmetry. --- Tangent bundle. --- Theorem. --- Vector bundle. --- Vector space. --- Vertical tangent. --- Winding number. --- Differential equations, Elliptic - Numerical solutions

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A survey, thorough and timely, of the singularities of two-dimensional normal complex analytic varieties, the volume summarizes the results obtained since Hirzebruch's thesis (1953) and presents new contributions. First, the singularity is resolved and shown to be classified by its resolution; then, resolutions are classed by the use of spaces with nilpotents; finally, the spaces with nilpotents are determined by means of the local ring structure of the singularity.

Algebraic geometry --- Analytic spaces --- SINGULARITIES (Mathematics) --- 512.76 --- Singularities (Mathematics) --- Geometry, Algebraic --- Spaces, Analytic --- Analytic functions --- Functions of several complex variables --- Birational geometry. Mappings etc. --- Analytic spaces. --- Singularities (Mathematics). --- 512.76 Birational geometry. Mappings etc. --- Birational geometry. Mappings etc --- Analytic function. --- Analytic set. --- Analytic space. --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Calculation. --- Chern class. --- Codimension. --- Coefficient. --- Cohomology. --- Compact Riemann surface. --- Complex manifold. --- Computation. --- Connected component (graph theory). --- Continuous function. --- Contradiction. --- Coordinate system. --- Corollary. --- Covering space. --- Dimension. --- Disjoint union. --- Divisor. --- Dual graph. --- Elliptic curve. --- Elliptic function. --- Embedding. --- Existential quantification. --- Factorization. --- Fiber bundle. --- Finite set. --- Formal power series. --- Hausdorff space. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Intersection (set theory). --- Intersection number (graph theory). --- Inverse limit. --- Irreducible component. --- Isolated singularity. --- Iteration. --- Lattice (group). --- Line bundle. --- Linear combination. --- Line–line intersection. --- Local coordinates. --- Local ring. --- Mathematical induction. --- Maximal ideal. --- Meromorphic function. --- Monic polynomial. --- Nilpotent. --- Normal bundle. --- Open set. --- Parameter. --- Plane curve. --- Pole (complex analysis). --- Power series. --- Presheaf (category theory). --- Projective line. --- Quadratic transformation. --- Quantity. --- Riemann surface. --- Riemann–Roch theorem. --- Several complex variables. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent space. --- Tensor algebra. --- Theorem. --- Topological space. --- Transition function. --- Two-dimensional space. --- Variable (mathematics). --- Zero divisor. --- Zero of a function. --- Zero set. --- Variétés complexes --- Espaces analytiques

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The intention of the authors is to examine the relationship between piecewise linear structure and differential structure: a relationship, they assert, that can be understood as a homotopy obstruction theory, and, hence, can be studied by using the traditional techniques of algebraic topology.Thus the book attacks the problem of existence and classification (up to isotopy) of differential structures compatible with a given combinatorial structure on a manifold. The problem is completely "solved" in the sense that it is reduced to standard problems of algebraic topology.The first part of the book is purely geometrical; it proves that every smoothing of the product of a manifold M and an interval is derived from an essentially unique smoothing of M. In the second part this result is used to translate the classification of smoothings into the problem of putting a linear structure on the tangent microbundle of M. This in turn is converted to the homotopy problem of classifying maps from M into a certain space PL/O. The set of equivalence classes of smoothings on M is given a natural abelian group structure.

Algebraic topology --- Piecewise linear topology --- Manifolds (Mathematics) --- Topologie linéaire par morceaux --- Variétés (Mathématiques) --- 515.16 --- PL topology --- Topology --- Geometry, Differential --- Topology of manifolds --- Piecewise linear topology. --- Manifolds (Mathematics). --- 515.16 Topology of manifolds --- Topologie linéaire par morceaux --- Variétés (Mathématiques) --- Affine transformation. --- Approximation. --- Associative property. --- Bijection. --- Bundle map. --- Classification theorem. --- Codimension. --- Coefficient. --- Cohomology. --- Commutative property. --- Computation. --- Convex cone. --- Convolution. --- Corollary. --- Counterexample. --- Diffeomorphism. --- Differentiable function. --- Differentiable manifold. --- Differential structure. --- Dimension. --- Direct proof. --- Division by zero. --- Embedding. --- Empty set. --- Equivalence class. --- Equivalence relation. --- Euclidean space. --- Existential quantification. --- Exponential map (Lie theory). --- Fiber bundle. --- Fibration. --- Functor. --- Grassmannian. --- H-space. --- Homeomorphism. --- Homotopy. --- Integral curve. --- Inverse problem. --- Isomorphism class. --- K0. --- Linearization. --- Manifold. --- Mathematical induction. --- Milnor conjecture. --- Natural transformation. --- Neighbourhood (mathematics). --- Normal bundle. --- Obstruction theory. --- Open set. --- Partition of unity. --- Piecewise linear. --- Polyhedron. --- Reflexive relation. --- Regular map (graph theory). --- Sheaf (mathematics). --- Smoothing. --- Smoothness. --- Special case. --- Submanifold. --- Tangent bundle. --- Tangent vector. --- Theorem. --- Topological manifold. --- Topological space. --- Topology. --- Transition function. --- Transitive relation. --- Vector bundle. --- Vector field. --- Variétés topologiques

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One of the great achievements of contemporary mathematics is the new understanding of four dimensions. Michael Freedman and Frank Quinn have been the principals in the geometric and topological development of this subject, proving the Poincar and Annulus conjectures respectively. Recognition for this work includes the award of the Fields Medal of the International Congress of Mathematicians to Freedman in 1986. In Topology of 4-Manifolds these authors have collaborated to give a complete and accessible account of the current state of knowledge in this field. The basic material has been considerably simplified from the original publications, and should be accessible to most graduate students. The advanced material goes well beyond the literature; nearly one-third of the book is new. This work is indispensable for any topologist whose work includes four dimensions. It is a valuable reference for geometers and physicists who need an awareness of the topological side of the field.Originally published in 1990.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.

Differential topology --- Four-manifolds (Topology) --- Trois-variétés (Topologie) --- Vier-menigvuldigheden (Topologie) --- 4-dimensional manifolds (Topology) --- 4-manifolds (Topology) --- Four dimensional manifolds (Topology) --- Manifolds, Four dimensional --- Low-dimensional topology --- Topological manifolds --- 4-manifold. --- Ambient isotopy. --- Annulus theorem. --- Automorphism. --- Baire category theorem. --- Bilinear form. --- Boundary (topology). --- CW complex. --- Category of manifolds. --- Central series. --- Characterization (mathematics). --- Cohomology. --- Commutative diagram. --- Commutative property. --- Commutator subgroup. --- Compactification (mathematics). --- Conformal geometry. --- Connected sum. --- Connectivity (graph theory). --- Cyclic group. --- Diagram (category theory). --- Diameter. --- Diffeomorphism. --- Differentiable manifold. --- Differential geometry. --- Dimension. --- Disk (mathematics). --- Duality (mathematics). --- Eigenvalues and eigenvectors. --- Embedding problem. --- Embedding. --- Equivariant map. --- Fiber bundle. --- Four-dimensional space. --- Fundamental group. --- General position. --- Geometry. --- H-cobordism. --- Handlebody. --- Hauptvermutung. --- Homeomorphism. --- Homology (mathematics). --- Homology sphere. --- Homomorphism. --- Homotopy group. --- Homotopy sphere. --- Homotopy. --- Hurewicz theorem. --- Hyperbolic geometry. --- Hyperbolic group. --- Hyperbolic manifold. --- Identity matrix. --- Intermediate value theorem. --- Intersection (set theory). --- Intersection curve. --- Intersection form (4-manifold). --- Intersection number (graph theory). --- Intersection number. --- J-homomorphism. --- Knot theory. --- Lefschetz duality. --- Line–line intersection. --- Manifold. --- Mapping cylinder. --- Mathematical induction. --- Metric space. --- Metrization theorem. --- Module (mathematics). --- Normal bundle. --- Parametrization. --- Parity (mathematics). --- Product topology. --- Pullback (differential geometry). --- Regular homotopy. --- Ring homomorphism. --- Rotation number. --- Seifert–van Kampen theorem. --- Sesquilinear form. --- Set (mathematics). --- Simply connected space. --- Smooth structure. --- Special case. --- Spin structure. --- Submanifold. --- Subset. --- Support (mathematics). --- Tangent bundle. --- Tangent space. --- Tensor product. --- Theorem. --- Topological category. --- Topological manifold. --- Transversal (geometry). --- Transversality (mathematics). --- Transversality theorem. --- Uniqueness theorem. --- Unit disk. --- Vector bundle. --- Whitehead torsion. --- Whitney disk.

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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