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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 theory of Riemann surfaces has a geometric and an analytic part. The former deals with the axiomatic definition of a Riemann surface, methods of construction, topological equivalence, and conformal mappings of one Riemann surface on another. The analytic part is concerned with the existence and properties of functions that have a special character connected with the conformal structure, for instance: subharmonic, harmonic, and analytic functions.Originally published in 1960.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.

515.16 --- 515.16 Topology of manifolds --- Topology of manifolds --- Riemann surfaces. --- Topology. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Surfaces, Riemann --- Functions --- Analytic function. --- Axiom of choice. --- Basis (linear algebra). --- Betti number. --- Big O notation. --- Bijection. --- Bilinear form. --- Bolzano–Weierstrass theorem. --- Boundary (topology). --- Boundary value problem. --- Bounded set (topological vector space). --- Branch point. --- Canonical basis. --- Cauchy sequence. --- Cauchy's integral formula. --- Characterization (mathematics). --- Coefficient. --- Commutator subgroup. --- Compact space. --- Compactification (mathematics). --- Conformal map. --- Connected space. --- Connectedness. --- Continuous function (set theory). --- Continuous function. --- Coset. --- Cross-cap. --- Dirichlet integral. --- Disjoint union. --- Elementary function. --- Elliptic surface. --- Exact differential. --- Existence theorem. --- Existential quantification. --- Extremal length. --- Family of sets. --- Finite intersection property. --- Finitely generated abelian group. --- Free group. --- Function (mathematics). --- Fundamental group. --- Green's function. --- Harmonic differential. --- Harmonic function. --- Harmonic measure. --- Heine–Borel theorem. --- Homeomorphism. --- Homology (mathematics). --- Ideal point. --- Infimum and supremum. --- Isolated point. --- Isolated singularity. --- Jordan curve theorem. --- Lebesgue integration. --- Limit point. --- Line segment. --- Linear independence. --- Linear map. --- Maximal set. --- Maximum principle. --- Meromorphic function. --- Metric space. --- Normal operator. --- Normal subgroup. --- Open set. --- Orientability. --- Orthogonal complement. --- Partition of unity. --- Point at infinity. --- Polyhedron. --- Positive harmonic function. --- Principal value. --- Projection (linear algebra). --- Projection (mathematics). --- Removable singularity. --- Riemann mapping theorem. --- Riemann surface. --- Semi-continuity. --- Sign (mathematics). --- Simplicial homology. --- Simply connected space. --- Singular homology. --- Skew-symmetric matrix. --- Special case. --- Subgroup. --- Subset. --- Summation. --- Support (mathematics). --- Taylor series. --- Theorem. --- Topological space. --- Triangle inequality. --- Uniform continuity. --- Uniformization theorem. --- Unit disk. --- Upper and lower bounds. --- Upper half-plane. --- Weyl's lemma (Laplace equation). --- Zorn's lemma.

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In this volume, the author covers profinite groups and their cohomology, Galois cohomology, and local class field theory, and concludes with a treatment of duality. His objective is to present effectively that body of material upon which all modern research in Diophantine geometry and higher arithmetic is based, and to do so in a manner that emphasizes the many interesting lines of inquiry leading from these foundations.

Group theory --- Finite groups --- Algebraic number theory --- 512.73 --- 512.66 --- Homology theory --- Number theory --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Groups, Finite --- Modules (Algebra) --- Cohomology theory of algebraic varieties and schemes --- Homological algebra --- 512.66 Homological algebra --- 512.73 Cohomology theory of algebraic varieties and schemes --- Groupes, Théorie des. --- Group theory. --- Homology theory. --- Finite groups. --- Algebraic number theory. --- Abelian group. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic extension. --- Algebraic geometry. --- Algebraic number field. --- Brauer group. --- Category of abelian groups. --- Category of sets. --- Characterization (mathematics). --- Class field theory. --- Cohomological dimension. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Composition series. --- Computation. --- Connected component (graph theory). --- Coset. --- Cup product. --- Dedekind domain. --- Degeneracy (mathematics). --- Diagram (category theory). --- Dimension (vector space). --- Diophantine geometry. --- Discrete group. --- Equivalence of categories. --- Exact sequence. --- Existential quantification. --- Explicit formula. --- Exponential function. --- Family of sets. --- Field extension. --- Finite group. --- Fundamental class. --- G-module. --- Galois cohomology. --- Galois extension. --- Galois group. --- Galois module. --- Galois theory. --- General topology. --- Geometry. --- Grothendieck topology. --- Group cohomology. --- Group extension. --- Group scheme. --- Hilbert symbol. --- Hopf algebra. --- Ideal (ring theory). --- Inequality (mathematics). --- Injective sheaf. --- Inner automorphism. --- Inverse limit. --- Kummer theory. --- Lie algebra. --- Linear independence. --- Local field. --- Mathematical induction. --- Mathematician. --- Mathematics. --- Module (mathematics). --- Morphism. --- Natural topology. --- Neighbourhood (mathematics). --- Normal extension. --- Normal subgroup. --- Number theory. --- P-adic number. --- P-group. --- Polynomial. --- Pontryagin duality. --- Power series. --- Prime number. --- Principal ideal. --- Profinite group. --- Quadratic reciprocity. --- Quotient group. --- Ring of integers. --- Sheaf (mathematics). --- Special case. --- Subcategory. --- Subgroup. --- Supernatural number. --- Sylow theorems. --- Tangent space. --- Theorem. --- Topological group. --- Topological property. --- Topological ring. --- Topological space. --- Topology. --- Torsion group. --- Torsion subgroup. --- Transcendence degree. --- Triviality (mathematics). --- Unique factorization domain. --- Variable (mathematics). --- Vector space. --- Groupes, Théorie des --- Nombres, Théorie des

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This book offers a systematic treatment--the first in book form--of the development and use of cohomological induction to construct unitary representations. George Mackey introduced induction in 1950 as a real analysis construction for passing from a unitary representation of a closed subgroup of a locally compact group to a unitary representation of the whole group. Later a parallel construction using complex analysis and its associated co-homology theories grew up as a result of work by Borel, Weil, Harish-Chandra, Bott, Langlands, Kostant, and Schmid. Cohomological induction, introduced by Zuckerman, is an algebraic analog that is technically more manageable than the complex-analysis construction and leads to a large repertory of irreducible unitary representations of reductive Lie groups. The book, which is accessible to students beyond the first year of graduate school, will interest mathematicians and physicists who want to learn about and take advantage of the algebraic side of the representation theory of Lie groups. Cohomological Induction and Unitary Representations develops the necessary background in representation theory and includes an introductory chapter of motivation, a thorough treatment of the "translation principle," and four appendices on algebra and analysis.

512.73 --- Harmonic analysis --- Homology theory --- Representations of groups --- Semisimple Lie groups --- Semi-simple Lie groups --- Lie groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Cohomology theory of algebraic varieties and schemes --- 512.73 Cohomology theory of algebraic varieties and schemes --- Lie algebras. --- Lie, Algèbres de. --- Semisimple Lie groups. --- Representations of groups. --- Homology theory. --- Harmonic analysis. --- Représentations d'algèbres de Lie --- Representations of Lie algebras --- Abelian category. --- Additive identity. --- Adjoint representation. --- Algebra homomorphism. --- Associative algebra. --- Associative property. --- Automorphic form. --- Automorphism. --- Banach space. --- Basis (linear algebra). --- Bilinear form. --- Cartan pair. --- Cartan subalgebra. --- Cartan subgroup. --- Cayley transform. --- Character theory. --- Classification theorem. --- Cohomology. --- Commutative property. --- Complexification (Lie group). --- Composition series. --- Conjugacy class. --- Conjugate transpose. --- Diagram (category theory). --- Dimension (vector space). --- Dirac delta function. --- Discrete series representation. --- Dolbeault cohomology. --- Eigenvalues and eigenvectors. --- Explicit formulae (L-function). --- Fubini's theorem. --- Functor. --- Gregg Zuckerman. --- Grothendieck group. --- Grothendieck spectral sequence. --- Haar measure. --- Hecke algebra. --- Hermite polynomials. --- Hermitian matrix. --- Hilbert space. --- Hilbert's basis theorem. --- Holomorphic function. --- Hopf algebra. --- Identity component. --- Induced representation. --- Infinitesimal character. --- Inner product space. --- Invariant subspace. --- Invariant theory. --- Inverse limit. --- Irreducible representation. --- Isomorphism class. --- Langlands classification. --- Langlands decomposition. --- Lexicographical order. --- Lie algebra. --- Linear extension. --- Linear independence. --- Mathematical induction. --- Matrix group. --- Module (mathematics). --- Monomial. --- Noetherian. --- Orthogonal transformation. --- Parabolic induction. --- Penrose transform. --- Projection (linear algebra). --- Reductive group. --- Representation theory. --- Semidirect product. --- Semisimple Lie algebra. --- Sesquilinear form. --- Sheaf cohomology. --- Skew-symmetric matrix. --- Special case. --- Spectral sequence. --- Stein manifold. --- Sub"ient. --- Subalgebra. --- Subcategory. --- Subgroup. --- Submanifold. --- Summation. --- Symmetric algebra. --- Symmetric space. --- Symmetrization. --- Tensor product. --- Theorem. --- Uniqueness theorem. --- Unitary group. --- Unitary operator. --- Unitary representation. --- Upper and lower bounds. --- Verma module. --- Weight (representation theory). --- Weyl character formula. --- Weyl group. --- Weyl's theorem. --- Zorn's lemma. --- Zuckerman functor.