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Study 79 contains a collection of papers presented at the Conference on Discontinuous Groups and Ricmann Surfaces at the University of Maryland, May 21-25, 1973. The papers, by leading authorities, deal mainly with Fuchsian and Kleinian groups, Teichmüller spaces, Jacobian varieties, and quasiconformal mappings. These topics are intertwined, representing a common meeting of algebra, geometry, and analysis.

Group theory --- Complex analysis --- Number theory --- RIEMANN SURFACES --- Discontinuous groups --- congresses --- Congresses --- Riemann surfaces --- Congresses. --- Groupes discontinus --- Combinatorial topology --- Functions of complex variables --- Surfaces, Riemann --- Functions --- Abelian variety. --- Adjunction (field theory). --- Affine space. --- Algebraic curve. --- Algebraic structure. --- Analytic function. --- Arithmetic genus. --- Automorphism. --- Bernhard Riemann. --- Boundary (topology). --- Cauchy sequence. --- Cauchy–Schwarz inequality. --- Cayley–Hamilton theorem. --- Closed geodesic. --- Combination. --- Commutative diagram. --- Commutator subgroup. --- Compact Riemann surface. --- Complex dimension. --- Complex manifold. --- Complex multiplication. --- Complex space. --- Complex torus. --- Congruence subgroup. --- Conjugacy class. --- Convex set. --- Cyclic group. --- Degeneracy (mathematics). --- Diagram (category theory). --- Diffeomorphism. --- Differential form. --- Dimension (vector space). --- Disjoint sets. --- E7 (mathematics). --- Endomorphism. --- Equation. --- Equivalence class. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- Finite group. --- Finitely generated group. --- Fuchsian group. --- Fundamental domain. --- Fundamental lemma (Langlands program). --- Fundamental polygon. --- Galois extension. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Homomorphism. --- Hurwitz's theorem (number theory). --- Inclusion map. --- Inequality (mathematics). --- Inner automorphism. --- Intersection (set theory). --- Irreducibility (mathematics). --- Isomorphism class. --- Isomorphism theorem. --- Jacobian variety. --- Jordan curve theorem. --- Kleinian group. --- Limit point. --- Mapping class group. --- Metric space. --- Monodromy. --- Monomorphism. --- Möbius transformation. --- Non-Euclidean geometry. --- Orthogonal trajectory. --- Permutation. --- Polynomial. --- Power series. --- Projective variety. --- Quadratic differential. --- Quadric. --- Quasi-projective variety. --- Quasiconformal mapping. --- Quotient space (topology). --- Rectangle. --- Riemann mapping theorem. --- Riemann surface. --- Schwarzian derivative. --- Simply connected space. --- Simultaneous equations. --- Special case. --- Subgroup. --- Subsequence. --- Surjective function. --- Symmetric space. --- Tangent space. --- Teichmüller space. --- Theorem. --- Topological space. --- Topology. --- Uniqueness theorem. --- Unit disk. --- Variable (mathematics). --- Winding number. --- Word problem (mathematics). --- RIEMANN SURFACES - congresses --- Discontinuous groups - Congresses --- Geometrie algebrique --- Fonctions d'une variable complexe --- Surfaces de riemann

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Resolution of singularities is a powerful and frequently used tool in algebraic geometry. In this book, János Kollár provides a comprehensive treatment of the characteristic 0 case. He describes more than a dozen proofs for curves, many based on the original papers of Newton, Riemann, and Noether. Kollár goes back to the original sources and presents them in a modern context. He addresses three methods for surfaces, and gives a self-contained and entirely elementary proof of a strong and functorial resolution in all dimensions. Based on a series of lectures at Princeton University and written in an informal yet lucid style, this book is aimed at readers who are interested in both the historical roots of the modern methods and in a simple and transparent proof of this important theorem.

Singularities (Mathematics) --- 512.761 --- Geometry, Algebraic --- Singularities. Singular points of algebraic varieties --- 512.761 Singularities. Singular points of algebraic varieties --- Adjunction formula. --- Algebraic closure. --- Algebraic geometry. --- Algebraic space. --- Algebraic surface. --- Algebraic variety. --- Approximation. --- Asymptotic analysis. --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Birational geometry. --- C0. --- Canonical singularity. --- Codimension. --- Cohomology. --- Commutative algebra. --- Complex analysis. --- Complex manifold. --- Computability. --- Continuous function. --- Coordinate system. --- Diagram (category theory). --- Differential geometry of surfaces. --- Dimension. --- Divisor. --- Du Val singularity. --- Dual graph. --- Embedding. --- Equation. --- Equivalence relation. --- Euclidean algorithm. --- Factorization. --- Functor. --- General position. --- Generic point. --- Geometric genus. --- Geometry. --- Hyperplane. --- Hypersurface. --- Integral domain. --- Intersection (set theory). --- Intersection number (graph theory). --- Intersection theory. --- Irreducible component. --- Isolated singularity. --- Laurent series. --- Line bundle. --- Linear space (geometry). --- Linear subspace. --- Mathematical induction. --- Mathematics. --- Maximal ideal. --- Morphism. --- Newton polygon. --- Noetherian ring. --- Noetherian. --- Open problem. --- Open set. --- P-adic number. --- Pairwise. --- Parametric equation. --- Partial derivative. --- Plane curve. --- Polynomial. --- Power series. --- Principal ideal. --- Principalization (algebra). --- Projective space. --- Projective variety. --- Proper morphism. --- Puiseux series. --- Quasi-projective variety. --- Rational function. --- Regular local ring. --- Resolution of singularities. --- Riemann surface. --- Ring theory. --- Ruler. --- Scientific notation. --- Sheaf (mathematics). --- Singularity theory. --- Smooth morphism. --- Smoothness. --- Special case. --- Subring. --- Summation. --- Surjective function. --- Tangent cone. --- Tangent space. --- Tangent. --- Taylor series. --- Theorem. --- Topology. --- Toric variety. --- Transversal (geometry). --- Variable (mathematics). --- Weierstrass preparation theorem. --- Weierstrass theorem. --- Zero set. --- Differential geometry. Global analysis

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The theory of Toeplitz operators has come to resemble more and more in recent years the classical theory of pseudodifferential operators. For instance, Toeplitz operators possess a symbolic calculus analogous to the usual symbolic calculus, and by symbolic means one can construct parametrices for Toeplitz operators and create new Toeplitz operators out of old ones by functional operations.If P is a self-adjoint pseudodifferential operator on a compact manifold with an elliptic symbol that is of order greater than zero, then it has a discrete spectrum. Also, it is well known that the asymptotic behavior of its eigenvalues is closely related to the behavior of the bicharacteristic flow generated by its symbol.It is natural to ask if similar results are true for Toeplitz operators. In the course of answering this question, the authors explore in depth the analogies between Toeplitz operators and pseudodifferential operators and show that both can be viewed as the "quantized" objects associated with functions on compact contact manifolds.

Operator theory --- Toeplitz operators --- Spectral theory (Mathematics) --- 517.984 --- Spectral theory of linear operators --- Toeplitz operators. --- Spectral theory (Mathematics). --- 517.984 Spectral theory of linear operators --- Operators, Toeplitz --- Linear operators --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Algebraic variety. --- Asymptotic analysis. --- Asymptotic expansion. --- Big O notation. --- Boundary value problem. --- Change of variables. --- Chern class. --- Codimension. --- Cohomology. --- Compact group. --- Complex manifold. --- Complex vector bundle. --- Connection form. --- Contact geometry. --- Corollary. --- Cotangent bundle. --- Curvature form. --- Diffeomorphism. --- Differentiable manifold. --- Dimensional analysis. --- Discrete spectrum. --- Eigenvalues and eigenvectors. --- Elaboration. --- Elliptic operator. --- Embedding. --- Equivalence class. --- Existential quantification. --- Exterior (topology). --- Fourier integral operator. --- Fourier transform. --- Hamiltonian vector field. --- Holomorphic function. --- Homogeneous function. --- Hypoelliptic operator. --- Integer. --- Integral curve. --- Integral transform. --- Invariant subspace. --- Lagrangian (field theory). --- Lagrangian. --- Limit point. --- Line bundle. --- Linear map. --- Mathematics. --- Metaplectic group. --- Natural number. --- Normal space. --- One-form. --- Open set. --- Operator (physics). --- Oscillatory integral. --- Parallel transport. --- Parameter. --- Parametrix. --- Periodic function. --- Polynomial. --- Projection (linear algebra). --- Projective variety. --- Pseudo-differential operator. --- Q.E.D. --- Quadratic form. --- Quantity. --- Quotient ring. --- Real number. --- Scientific notation. --- Self-adjoint. --- Smoothness. --- Spectral theorem. --- Spectral theory. --- Square root. --- Submanifold. --- Summation. --- Support (mathematics). --- Symplectic geometry. --- Symplectic group. --- Symplectic manifold. --- Symplectic vector space. --- Tangent space. --- Theorem. --- Todd class. --- Toeplitz algebra. --- Toeplitz matrix. --- Toeplitz operator. --- Trace formula. --- Transversal (geometry). --- Trigonometric functions. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume form. --- Wave front set. --- Opérateurs pseudo-différentiels

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The theory of D-modules deals with the algebraic aspects of differential equations. These are particularly interesting on homogeneous manifolds, since the infinitesimal action of a Lie algebra consists of differential operators. Hence, it is possible to attach geometric invariants, like the support and the characteristic variety, to representations of Lie groups. By considering D-modules on flag varieties, one obtains a simple classification of all irreducible admissible representations of reductive Lie groups. On the other hand, it is natural to study the representations realized by functions on pseudo-Riemannian symmetric spaces, i.e., spherical representations. The problem is then to describe the spherical representations among all irreducible ones, and to compute their multiplicities. This is the goal of this work, achieved fairly completely at least for the discrete series representations of reductive symmetric spaces. The book provides a general introduction to the theory of D-modules on flag varieties, and it describes spherical D-modules in terms of a cohomological formula. Using microlocalization of representations, the author derives a criterion for irreducibility. The relation between multiplicities and singularities is also discussed at length.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.

Differentiable manifolds. --- D-modules. --- Representations of groups. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Modules (Algebra) --- Differential manifolds --- Manifolds (Mathematics) --- Affine space. --- Algebraic cycle. --- Algebraic element. --- Analytic function. --- Annihilator (ring theory). --- Automorphism. --- Banach space. --- Base change. --- Big O notation. --- Bijection. --- Bilinear form. --- Borel subgroup. --- Cartan subalgebra. --- Cofibration. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Commutator subgroup. --- Complexification (Lie group). --- Conjugacy class. --- Coproduct. --- Coset. --- Cotangent space. --- D-module. --- Derived category. --- Diagram (category theory). --- Differential operator. --- Dimension (vector space). --- Direct image functor. --- Discrete series representation. --- Disk (mathematics). --- Dot product. --- Double coset. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Euler operator. --- Existential quantification. --- Fibration. --- Function space. --- Functor. --- G-module. --- Gelfand pair. --- Generic point. --- Hilbert space. --- Holomorphic function. --- Homomorphism. --- Hyperfunction. --- Ideal (ring theory). --- Infinitesimal character. --- Inner automorphism. --- Invertible sheaf. --- Irreducibility (mathematics). --- Irreducible representation. --- Levi decomposition. --- Lie algebra. --- Line bundle. --- Linear algebraic group. --- Linear space (geometry). --- Manifold. --- Maximal compact subgroup. --- Maximal torus. --- Metric space. --- Module (mathematics). --- Moment map. --- Morphism. --- Noetherian ring. --- Open set. --- Presheaf (category theory). --- Principal series representation. --- Projective line. --- Projective object. --- Projective space. --- Projective variety. --- Reductive group. --- Riemannian geometry. --- Riemann–Hilbert correspondence. --- Right inverse. --- Ring (mathematics). --- Root system. --- Satake diagram. --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Sphere. --- Square-integrable function. --- Sub"ient. --- Subalgebra. --- Subcategory. --- Subgroup. --- Summation. --- Surjective function. --- Symmetric space. --- Symplectic geometry. --- Tensor product. --- Theorem. --- Triangular matrix. --- Vector bundle. --- Volume form. --- Weyl group.

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The description for this book, Riemann Surfaces Related Topics (AM-97), Volume 97: Proceedings of the 1978 Stony Brook Conference. (AM-97), will be forthcoming.

Geometry --- Riemann surfaces --- -517.54 --- Surfaces, Riemann --- Functions --- Congresses --- Conformal mapping and geometric problems in the theory of functions of a complex variable. Analytic functions and their generalizations --- 517.54 Conformal mapping and geometric problems in the theory of functions of a complex variable. Analytic functions and their generalizations --- 517.54 --- Riemann, Surfaces de --- Abstract simplicial complex. --- Affine transformation. --- Algebraic curve. --- Algebraic element. --- Algebraic equation. --- Algebraic surface. --- Analytic function. --- Analytic torsion. --- Automorphic form. --- Automorphic function. --- Automorphism. --- Banach space. --- Basis (linear algebra). --- Boundary (topology). --- Bounded set (topological vector space). --- Cohomology ring. --- Cohomology. --- Commutative property. --- Commutator subgroup. --- Compact Riemann surface. --- Complex analysis. --- Complex manifold. --- Conformal geometry. --- Conformal map. --- Conjugacy class. --- Covering space. --- Diagram (category theory). --- Dimension (vector space). --- Divisor (algebraic geometry). --- Divisor. --- Eigenvalues and eigenvectors. --- Equivalence class. --- Equivalence relation. --- Ergodic theory. --- Existential quantification. --- Foliation. --- Fuchsian group. --- Fundamental domain. --- Fundamental group. --- Fundamental polygon. --- Geodesic. --- Geometric function theory. --- Group homomorphism. --- H-cobordism. --- Hausdorff measure. --- Holomorphic function. --- Homeomorphism. --- Homomorphism. --- Homotopy. --- Hyperbolic 3-manifold. --- Hyperbolic manifold. --- Hyperbolic space. --- Infimum and supremum. --- Injective module. --- Interior (topology). --- Intersection form (4-manifold). --- Isometry. --- Isomorphism class. --- Jordan curve theorem. --- Kleinian group. --- Kähler manifold. --- Limit point. --- Limit set. --- Manifold. --- Meromorphic function. --- Metric space. --- Mostow rigidity theorem. --- Möbius transformation. --- Poincaré conjecture. --- Pole (complex analysis). --- Polynomial. --- Product topology. --- Projective variety. --- Quadratic differential. --- Quasi-isometry. --- Quasiconformal mapping. --- Quotient space (topology). --- Radon–Nikodym theorem. --- Ricci curvature. --- Riemann mapping theorem. --- Riemann sphere. --- Riemann surface. --- Riemannian geometry. --- Riemannian manifold. --- Schwarzian derivative. --- Strictly convex space. --- Subgroup. --- Submanifold. --- Surjective function. --- Tangent space. --- Teichmüller space. --- Theorem. --- Topological conjugacy. --- Topological space. --- Topology. --- Uniformization theorem. --- Uniformization. --- Uniqueness theorem. --- Unit disk. --- Vector bundle.