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This monograph aims to give a self-contained introduction into the whole field of topological analysis: Requiring essentially only basic knowledge of elementary calculus and linear algebra, it provides all required background from topology, analysis, linear and nonlinear functional analysis, and multivalued maps, containing even basic topics like separation axioms, inverse and implicit function theorems, the Hahn-Banach theorem, Banach manifolds, or the most important concepts of continuity of multivalued maps. Thus, it can be used as additional material in basic courses on such topics. The main intention, however, is to provide also additional information on some fine points which are usually not discussed in such introductory courses. The selection of the topics is mainly motivated by the requirements for degree theory which is presented in various variants, starting from the elementary Brouwer degree (in Euclidean spaces and on manifolds) with several of its famous classical consequences, up to a general degree theory for function triples which applies for a large class of problems in a natural manner. Although it has been known to specialists that, in principle, such a general degree theory must exist, this is the first monograph in which the corresponding theory is developed in detail.

Topological degree. --- Topological spaces. --- Fredholm operators. --- Algebraic topology. --- Topology --- Operators, Fredholm --- Linear operators --- Spaces, Topological --- Degree, Topological --- Degree (Topology) --- Degree theory --- Algebraic topology --- Analysis. --- Banach Manifold. --- Fredholm. --- Hahn-Banach Theorem. --- Implicit Function Theorem. --- Inverse Function Theorem. --- Linear Functional Analysis. --- Multivalued Map. --- Nonlinear Functional Analysis. --- Nonlinear Inclusion. --- Separation Axiom. --- Topology. --- Triple Degree.

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Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence. This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue. Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory. Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.

Differential geometry. Global analysis --- Bifurcation theory. --- Differential equations, Nonlinear --- Stability --- Numerical solutions --- Addition. --- Algebraic equation. --- Analytic function. --- Analytic manifold. --- Atmospheric pressure. --- Banach space. --- Bernhard Riemann. --- Bifurcation diagram. --- Boundary value problem. --- Bounded operator. --- Bounded set (topological vector space). --- Boundedness. --- Canonical form. --- Cartesian coordinate system. --- Codimension. --- Compact operator. --- Complex analysis. --- Complex conjugate. --- Complex number. --- Connected space. --- Coordinate system. --- Corollary. --- Curvature. --- Derivative. --- Diagram (category theory). --- Differentiable function. --- Differentiable manifold. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Eigenvalues and eigenvectors. --- Elliptic integral. --- Embedding. --- Equation. --- Euclidean division. --- Euler equations (fluid dynamics). --- Existential quantification. --- First principle. --- Fredholm operator. --- Froude number. --- Functional analysis. --- Hilbert space. --- Homeomorphism. --- Implicit function theorem. --- Integer. --- Linear algebra. --- Linear function. --- Linear independence. --- Linear map. --- Linear programming. --- Linear space (geometry). --- Linear subspace. --- Linearity. --- Linearization. --- Metric space. --- Morse theory. --- Multilinear form. --- N0. --- Natural number. --- Neumann series. --- Nonlinear functional analysis. --- Nonlinear system. --- Numerical analysis. --- Open mapping theorem (complex analysis). --- Operator (physics). --- Ordinary differential equation. --- Parameter. --- Parametrization. --- Partial differential equation. --- Permutation group. --- Permutation. --- Polynomial. --- Power series. --- Prime number. --- Proportionality (mathematics). --- Pseudo-differential operator. --- Puiseux series. --- Quantity. --- Real number. --- Resultant. --- Singularity theory. --- Skew-symmetric matrix. --- Smoothness. --- Solution set. --- Special case. --- Standard basis. --- Sturm–Liouville theory. --- Subset. --- Symmetric bilinear form. --- Symmetric group. --- Taylor series. --- Taylor's theorem. --- Theorem. --- Total derivative. --- Two-dimensional space. --- Union (set theory). --- Variable (mathematics). --- Vector space. --- Zero of a function.

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The recent introduction of the Seiberg-Witten invariants of smooth four-manifolds has revolutionized the study of those manifolds. The invariants are gauge-theoretic in nature and are close cousins of the much-studied SU(2)-invariants defined over fifteen years ago by Donaldson. On a practical level, the new invariants have proved to be more powerful and have led to a vast generalization of earlier results. This book is an introduction to the Seiberg-Witten invariants. The work begins with a review of the classical material on Spin c structures and their associated Dirac operators. Next comes a discussion of the Seiberg-Witten equations, which is set in the context of nonlinear elliptic operators on an appropriate infinite dimensional space of configurations. It is demonstrated that the space of solutions to these equations, called the Seiberg-Witten moduli space, is finite dimensional, and its dimension is then computed. In contrast to the SU(2)-case, the Seiberg-Witten moduli spaces are shown to be compact. The Seiberg-Witten invariant is then essentially the homology class in the space of configurations represented by the Seiberg-Witten moduli space. The last chapter gives a flavor for the applications of these new invariants by computing the invariants for most Kahler surfaces and then deriving some basic toological consequences for these surfaces.

Four-manifolds (Topology) --- Seiberg-Witten invariants. --- Mathematical physics. --- Physical mathematics --- Physics --- Invariants --- 4-dimensional manifolds (Topology) --- 4-manifolds (Topology) --- Four dimensional manifolds (Topology) --- Manifolds, Four dimensional --- Low-dimensional topology --- Topological manifolds --- Mathematics --- Affine space. --- Affine transformation. --- Algebra bundle. --- Algebraic surface. --- Almost complex manifold. --- Automorphism. --- Banach space. --- Clifford algebra. --- Cohomology. --- Cokernel. --- Complex dimension. --- Complex manifold. --- Complex plane. --- Complex projective space. --- Complex vector bundle. --- Complexification (Lie group). --- Computation. --- Configuration space. --- Conjugate transpose. --- Covariant derivative. --- Curvature form. --- Curvature. --- Differentiable manifold. --- Differential topology. --- Dimension (vector space). --- Dirac equation. --- Dirac operator. --- Division algebra. --- Donaldson theory. --- Duality (mathematics). --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Elliptic surface. --- Equation. --- Fiber bundle. --- Frenet–Serret formulas. --- Gauge fixing. --- Gauge theory. --- Gaussian curvature. --- Geometry. --- Group homomorphism. --- Hilbert space. --- Hodge index theorem. --- Homology (mathematics). --- Homotopy. --- Identity (mathematics). --- Implicit function theorem. --- Intersection form (4-manifold). --- Inverse function theorem. --- Isomorphism class. --- K3 surface. --- Kähler manifold. --- Levi-Civita connection. --- Lie algebra. --- Line bundle. --- Linear map. --- Linear space (geometry). --- Linearization. --- Manifold. --- Mathematical induction. --- Moduli space. --- Multiplication theorem. --- Neighbourhood (mathematics). --- One-form. --- Open set. --- Orientability. --- Orthonormal basis. --- Parameter space. --- Parametric equation. --- Parity (mathematics). --- Partial derivative. --- Principal bundle. --- Projection (linear algebra). --- Pullback (category theory). --- Quadratic form. --- Quaternion algebra. --- Quotient space (topology). --- Riemann surface. --- Riemannian manifold. --- Sard's theorem. --- Sign (mathematics). --- Sobolev space. --- Spin group. --- Spin representation. --- Spin structure. --- Spinor field. --- Subgroup. --- Submanifold. --- Surjective function. --- Symplectic geometry. --- Symplectic manifold. --- Tangent bundle. --- Tangent space. --- Tensor product. --- Theorem. --- Three-dimensional space (mathematics). --- Trace (linear algebra). --- Transversality (mathematics). --- Two-form. --- Zariski tangent space.

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Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics. Its central concern is the structure of an infinitely renormalizable quadratic polynomial f(z) = z2 + c. As discovered by Feigenbaum, such a mapping exhibits a repetition of form at infinitely many scales. Drawing on universal estimates in hyperbolic geometry, this work gives an analysis of the limiting forms that can occur and develops a rigidity criterion for the polynomial f. This criterion supports general conjectures about the behavior of rational maps and the structure of the Mandelbrot set. The course of the main argument entails many facets of modern complex dynamics. Included are foundational results in geometric function theory, quasiconformal mappings, and hyperbolic geometry. Most of the tools are discussed in the setting of general polynomials and rational maps.

Renormalization (Physics) --- Polynomials. --- Dynamics. --- Mathematical physics. --- Analytic function. --- Attractor. --- Automorphism. --- Bernhard Riemann. --- Bounded set. --- Branched covering. --- Cantor set. --- Cardioid. --- Chain rule. --- Coefficient. --- Combinatorics. --- Complex manifold. --- Complex plane. --- Complex torus. --- Conformal geometry. --- Conformal map. --- Conjecture. --- Connected space. --- Covering space. --- Cyclic group. --- Degeneracy (mathematics). --- Dense set. --- Diagram (category theory). --- Diameter. --- Differential geometry of surfaces. --- Dihedral group. --- Dimension (vector space). --- Dimension. --- Disjoint sets. --- Disk (mathematics). --- Dynamical system. --- Endomorphism. --- Equivalence class. --- Equivalence relation. --- Ergodic theory. --- Euler characteristic. --- Filled Julia set. --- Geometric function theory. --- Geometry. --- Hausdorff dimension. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Hyperbolic geometry. --- Implicit function theorem. --- Injective function. --- Integer matrix. --- Interval (mathematics). --- Inverse limit. --- Julia set. --- Kleinian group. --- Limit point. --- Limit set. --- Linear map. --- Mandelbrot set. --- Manifold. --- Markov partition. --- Mathematical induction. --- Maxima and minima. --- Measure (mathematics). --- Moduli (physics). --- Monic polynomial. --- Montel's theorem. --- Möbius transformation. --- Natural number. --- Open set. --- Orbifold. --- Periodic point. --- Permutation. --- Point at infinity. --- Pole (complex analysis). --- Polynomial. --- Proper map. --- Quadratic differential. --- Quadratic function. --- Quadratic. --- Quasi-isometry. --- Quasiconformal mapping. --- Quotient space (topology). --- Removable singularity. --- Renormalization. --- Riemann mapping theorem. --- Riemann sphere. --- Riemann surface. --- Rigidity theory (physics). --- Scalar (physics). --- Schwarz lemma. --- Scientific notation. --- Special case. --- Structural stability. --- Subgroup. --- Subsequence. --- Symbolic dynamics. --- Tangent space. --- Theorem. --- Uniformization theorem. --- Uniformization. --- Union (set theory). --- Unit disk. --- Upper and lower bounds.

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This book builds upon the revolutionary discovery made in 1974 that when one passes from function f to a function J of paths joining two points A1≠A1 the connectivities R1 of the domain of f can be replaced by connectivities R1 over Q, common to the pathwise components of a basic Frechet space of classes of equivalent curves joining A1 to A1. The connectivities R1, termed "Frechet numbers," are proved independent of the choice of A1 ≠ A1, and of a replacement of Mn by any differential manifold homeomorphic to Mn.Originally published in 1976.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.

Differentiable manifolds. --- Global analysis (Mathematics) --- Calculus of variations. --- Algebraic topology. --- Analytic function. --- Arc length. --- Axiom. --- Bernhard Riemann. --- Boundary value problem. --- Cartesian coordinate system. --- Coefficient. --- Compact space. --- Computation. --- Conjugate points. --- Connectivity (graph theory). --- Continuous function. --- Corollary. --- Countable set. --- Counting. --- Cramer's rule. --- Curve. --- Deformation theory. --- Degeneracy (mathematics). --- Derivative. --- Diffeomorphism. --- Differentiable manifold. --- Differential equation. --- Differential geometry. --- Differential structure. --- Dimension. --- Domain of a function. --- Eilenberg. --- Einstein notation. --- Equation. --- Euclidean space. --- Euler characteristic. --- Euler equations (fluid dynamics). --- Euler integral. --- Existence theorem. --- Existential quantification. --- Exotic sphere. --- Family of curves. --- Finite set. --- First variation. --- Geometry. --- Global analysis. --- Homeomorphism. --- Homology (mathematics). --- Homotopy. --- Implicit function theorem. --- Inference. --- Integer. --- Intersection (set theory). --- Interval (mathematics). --- Invertible matrix. --- Jacobian matrix and determinant. --- Lagrange multiplier. --- Linear combination. --- Linear map. --- Line–line intersection. --- Mathematical proof. --- Maximal set. --- Metric space. --- N-sphere. --- Neighbourhood (mathematics). --- Null vector. --- Open set. --- Pairwise. --- Parameter. --- Parametric equation. --- Parametrization. --- Partial derivative. --- Partial function. --- Phase space. --- Positive definiteness. --- Projective plane. --- Quadratic form. --- Quadratic. --- Rate of convergence. --- Rational number. --- Real variable. --- Resultant. --- Riemannian manifold. --- Scientific notation. --- Sign (mathematics). --- Special case. --- Sturm separation theorem. --- Submanifold. --- Subsequence. --- Subset. --- Taylor's theorem. --- Tensor algebra. --- Theorem. --- Theory. --- Topological manifold. --- Topological space. --- Topology. --- Tuple. --- Unit vector. --- Variable (mathematics). --- Variational analysis. --- Weierstrass function. --- Without loss of generality.