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Part explanation of important recent work, and part introduction to some of the techniques of modern partial differential equations, this monograph is a self-contained exposition of the Neumann problem for the Cauchy-Riemann complex and certain of its applications. The authors prove the main existence and regularity theorems in detail, assuming only a knowledge of the basic theory of differentiable manifolds and operators on Hilbert space. They discuss applications to the theory of several complex variables, examine the associated complex on the boundary, and outline other techniques relevant to these problems. In an appendix they develop the functional analysis of differential operators in terms of Sobolev spaces, to the extent it is required for the monograph.

Functional analysis --- Neumann problem --- Differential operators --- Complex manifolds --- Complex manifolds. --- Differential operators. --- Neumann problem. --- Differential equations, Partial --- Équations aux dérivées partielles --- Analytic spaces --- Manifolds (Mathematics) --- Operators, Differential --- Differential equations --- Operator theory --- Boundary value problems --- A priori estimate. --- Almost complex manifold. --- Analytic function. --- Apply. --- Approximation. --- Bernhard Riemann. --- Boundary value problem. --- Calculation. --- Cauchy–Riemann equations. --- Cohomology. --- Compact space. --- Complex analysis. --- Complex manifold. --- Coordinate system. --- Corollary. --- Derivative. --- Differentiable manifold. --- Differential equation. --- Differential form. --- Differential operator. --- Dimension (vector space). --- Dirichlet boundary condition. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Equation. --- Estimation. --- Euclidean space. --- Existence theorem. --- Exterior (topology). --- Finite difference. --- Fourier analysis. --- Fourier transform. --- Frobenius theorem (differential topology). --- Functional analysis. --- Hilbert space. --- Hodge theory. --- Holomorphic function. --- Holomorphic vector bundle. --- Irreducible representation. --- Line segment. --- Linear programming. --- Local coordinates. --- Lp space. --- Manifold. --- Monograph. --- Multi-index notation. --- Nonlinear system. --- Operator (physics). --- Overdetermined system. --- Partial differential equation. --- Partition of unity. --- Potential theory. --- Power series. --- Pseudo-differential operator. --- Pseudoconvexity. --- Pseudogroup. --- Pullback. --- Regularity theorem. --- Remainder. --- Scientific notation. --- Several complex variables. --- Sheaf (mathematics). --- Smoothness. --- Sobolev space. --- Special case. --- Statistical significance. --- Sturm–Liouville theory. --- Submanifold. --- Tangent bundle. --- Theorem. --- Uniform norm. --- Vector field. --- Weight function. --- Operators in hilbert space --- Équations aux dérivées partielles

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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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Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic. Ravenel's first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.

Homotopie --- Homotopy theory --- Homotopy theory. --- Deformations, Continuous --- Topology --- Abelian category. --- Abelian group. --- Adams spectral sequence. --- Additive category. --- Affine space. --- Algebra homomorphism. --- Algebraic closure. --- Algebraic structure. --- Algebraic topology (object). --- Algebraic topology. --- Algebraic variety. --- Algebraically closed field. --- Atiyah–Hirzebruch spectral sequence. --- Automorphism. --- Boolean algebra (structure). --- CW complex. --- Canonical map. --- Cantor set. --- Category of topological spaces. --- Category theory. --- Classification theorem. --- Classifying space. --- Cohomology operation. --- Cohomology. --- Cokernel. --- Commutative algebra. --- Commutative ring. --- Complex projective space. --- Complex vector bundle. --- Computation. --- Conjecture. --- Conjugacy class. --- Continuous function. --- Contractible space. --- Coproduct. --- Differentiable manifold. --- Disjoint union. --- Division algebra. --- Equation. --- Explicit formulae (L-function). --- Functor. --- G-module. --- Groupoid. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Hopf algebra. --- Hurewicz theorem. --- Inclusion map. --- Infinite product. --- Integer. --- Inverse limit. --- Irreducible representation. --- Isomorphism class. --- K-theory. --- Loop space. --- Mapping cone (homological algebra). --- Mathematical induction. --- Modular representation theory. --- Module (mathematics). --- Monomorphism. --- Moore space. --- Morava K-theory. --- Morphism. --- N-sphere. --- Noetherian ring. --- Noetherian. --- Noncommutative ring. --- Number theory. --- P-adic number. --- Piecewise linear manifold. --- Polynomial ring. --- Polynomial. --- Power series. --- Prime number. --- Principal ideal domain. --- Profinite group. --- Reduced homology. --- Ring (mathematics). --- Ring homomorphism. --- Ring spectrum. --- Simplicial complex. --- Simply connected space. --- Smash product. --- Special case. --- Spectral sequence. --- Steenrod algebra. --- Sub"ient. --- Subalgebra. --- Subcategory. --- Subring. --- Symmetric group. --- Tensor product. --- Theorem. --- Topological space. --- Topology. --- Vector bundle. --- Zariski topology.

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For hundreds of years, the study of elliptic curves has played a central role in mathematics. The past century in particular has seen huge progress in this study, from Mordell's theorem in 1922 to the work of Wiles and Taylor-Wiles in 1994. Nonetheless, there remain many fundamental questions where we do not even know what sort of answers to expect. This book explores two of them: What is the average rank of elliptic curves, and how does the rank vary in various kinds of families of elliptic curves? Nicholas Katz answers these questions for families of ''big'' twists of elliptic curves in the function field case (with a growing constant field). The monodromy-theoretic methods he develops turn out to apply, still in the function field case, equally well to families of big twists of objects of all sorts, not just to elliptic curves. The leisurely, lucid introduction gives the reader a clear picture of what is known and what is unknown at present, and situates the problems solved in this book within the broader context of the overall study of elliptic curves. The book's technical core makes use of, and explains, various advanced topics ranging from recent results in finite group theory to the machinery of l-adic cohomology and monodromy. Twisted L-Functions and Monodromy is essential reading for anyone interested in number theory and algebraic geometry.

L-functions. --- Monodromy groups. --- Functions, L --- -L-functions. --- Group theory --- -Number theory --- L-functions --- Monodromy groups --- Abelian variety. --- Absolute continuity. --- Addition. --- Affine space. --- Algebraically closed field. --- Ambient space. --- Average. --- Betti number. --- Birch and Swinnerton-Dyer conjecture. --- Blowing up. --- Codimension. --- Coefficient. --- Computation. --- Conjecture. --- Conjugacy class. --- Convolution. --- Critical value. --- Differential geometry of surfaces. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Divisor (algebraic geometry). --- Divisor. --- Eigenvalues and eigenvectors. --- Elliptic curve. --- Equation. --- Equidistribution theorem. --- Existential quantification. --- Factorization. --- Finite field. --- Finite group. --- Finite set. --- Flat map. --- Fourier transform. --- Function field. --- Functional equation. --- Goursat's lemma. --- Ground field. --- Group representation. --- Hyperplane. --- Hypersurface. --- Integer matrix. --- Integer. --- Irreducible component. --- Irreducible polynomial. --- Irreducible representation. --- J-invariant. --- K3 surface. --- L-function. --- Lebesgue measure. --- Lefschetz pencil. --- Level of measurement. --- Lie algebra. --- Limit superior and limit inferior. --- Minimal polynomial (field theory). --- Modular form. --- Monodromy. --- Morphism. --- Numerical analysis. --- Orthogonal group. --- Percentage. --- Polynomial. --- Prime number. --- Probability measure. --- Quadratic function. --- Quantity. --- Quotient space (topology). --- Representation theory. --- Residue field. --- Riemann hypothesis. --- Root of unity. --- Scalar (physics). --- Set (mathematics). --- Sheaf (mathematics). --- Subgroup. --- Summation. --- Symmetric group. --- System of imprimitivity. --- Theorem. --- Trivial representation. --- Zariski topology.

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This book studies the interplay between the geometry and topology of locally symmetric spaces, and the arithmetic aspects of the special values of L-functions.The authors study the cohomology of locally symmetric spaces for GL(N) where the cohomology groups are with coefficients in a local system attached to a finite-dimensional algebraic representation of GL(N). The image of the global cohomology in the cohomology of the Borel-Serre boundary is called Eisenstein cohomology, since at a transcendental level the cohomology classes may be described in terms of Eisenstein series and induced representations. However, because the groups are sheaf-theoretically defined, one can control their rationality and even integrality properties. A celebrated theorem by Langlands describes the constant term of an Eisenstein series in terms of automorphic L-functions. A cohomological interpretation of this theorem in terms of maps in Eisenstein cohomology allows the authors to study the rationality properties of the special values of Rankin-Selberg L-functions for GL(n) x GL(m), where n + m = N. The authors carry through the entire program with an eye toward generalizations.This book should be of interest to advanced graduate students and researchers interested in number theory, automorphic forms, representation theory, and the cohomology of arithmetic groups.

Shimura varieties. --- Cohomology operations. --- Number theory. --- Arithmetic groups. --- L-functions. --- Functions, L --- -Number theory --- Group theory --- Number study --- Numbers, Theory of --- Algebra --- Operations (Algebraic topology) --- Algebraic topology --- Varieties, Shimura --- Arithmetical algebraic geometry --- Addition. --- Adele ring. --- Algebraic group. --- Algebraic number theory. --- Arithmetic group. --- Automorphic form. --- Base change. --- Basis (linear algebra). --- Bearing (navigation). --- Borel subgroup. --- Calculation. --- Category of groups. --- Coefficient. --- Cohomology. --- Combination. --- Commutative ring. --- Compact group. --- Computation. --- Conjecture. --- Constant term. --- Corollary. --- Covering space. --- Critical value. --- Diagram (category theory). --- Dimension. --- Dirichlet character. --- Discrete series representation. --- Discrete spectrum. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Elaboration. --- Embedding. --- Euler product. --- Field extension. --- Field of fractions. --- Free module. --- Freydoon Shahidi. --- Function field. --- Functor. --- Galois group. --- Ground field. --- Group (mathematics). --- Group scheme. --- Harish-Chandra. --- Hecke L-function. --- Hecke character. --- Hecke operator. --- Hereditary property. --- Induced representation. --- Irreducible representation. --- K0. --- L-function. --- Langlands dual group. --- Level structure. --- Lie algebra cohomology. --- Lie algebra. --- Lie group. --- Linear combination. --- Linear map. --- Local system. --- Maximal torus. --- Modular form. --- Modular symbol. --- Module (mathematics). --- Monograph. --- N0. --- National Science Foundation. --- Natural number. --- Natural transformation. --- Nilradical. --- Permutation. --- Prime number. --- Quantity. --- Rational number. --- Reductive group. --- Requirement. --- Ring of integers. --- Root of unity. --- SL2(R). --- Scalar (physics). --- Sheaf (mathematics). --- Special case. --- Spectral sequence. --- Standard L-function. --- Subgroup. --- Subset. --- Summation. --- Tensor product. --- Theorem. --- Theory. --- Triangular matrix. --- Triviality (mathematics). --- Two-dimensional space. --- Unitary group. --- Vector space. --- W0. --- Weyl group.