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In 1970, Phillip Griffiths envisioned that points at infinity could be added to the classifying space D of polarized Hodge structures. In this book, Kazuya Kato and Sampei Usui realize this dream by creating a logarithmic Hodge theory. They use the logarithmic structures begun by Fontaine-Illusie to revive nilpotent orbits as a logarithmic Hodge structure. The book focuses on two principal topics. First, Kato and Usui construct the fine moduli space of polarized logarithmic Hodge structures with additional structures. Even for a Hermitian symmetric domain D, the present theory is a refinement of the toroidal compactifications by Mumford et al. For general D, fine moduli spaces may have slits caused by Griffiths transversality at the boundary and be no longer locally compact. Second, Kato and Usui construct eight enlargements of D and describe their relations by a fundamental diagram, where four of these enlargements live in the Hodge theoretic area and the other four live in the algebra-group theoretic area. These two areas are connected by a continuous map given by the SL(2)-orbit theorem of Cattani-Kaplan-Schmid. This diagram is used for the construction in the first topic.

Hodge theory. --- Logarithms. --- Logs (Logarithms) --- Algebra --- Complex manifolds --- Differentiable manifolds --- Geometry, Algebraic --- Homology theory --- Algebraic group. --- Algebraic variety. --- Analytic manifold. --- Analytic space. --- Annulus (mathematics). --- Arithmetic group. --- Atlas (topology). --- Canonical map. --- Classifying space. --- Coefficient. --- Cohomology. --- Compactification (mathematics). --- Complex manifold. --- Complex number. --- Congruence subgroup. --- Conjecture. --- Connected component (graph theory). --- Continuous function. --- Convex cone. --- Degeneracy (mathematics). --- Diagram (category theory). --- Differential form. --- Direct image functor. --- Divisor. --- Elliptic curve. --- Equivalence class. --- Existential quantification. --- Finite set. --- Functor. --- Geometry. --- Hodge structure. --- Homeomorphism. --- Homomorphism. --- Inverse function. --- Iwasawa decomposition. --- Local homeomorphism. --- Local ring. --- Local system. --- Logarithmic. --- Maximal compact subgroup. --- Modular curve. --- Modular form. --- Moduli space. --- Monodromy. --- Monoid. --- Morphism. --- Natural number. --- Nilpotent orbit. --- Nilpotent. --- Open problem. --- Open set. --- P-adic Hodge theory. --- P-adic number. --- Point at infinity. --- Proper morphism. --- Pullback (category theory). --- Quotient space (topology). --- Rational number. --- Relative interior. --- Ring (mathematics). --- Ring homomorphism. --- Scientific notation. --- Set (mathematics). --- Sheaf (mathematics). --- Smooth morphism. --- Special case. --- Strong topology. --- Subgroup. --- Subobject. --- Subset. --- Surjective function. --- Tangent bundle. --- Taylor series. --- Theorem. --- Topological space. --- Topology. --- Transversality (mathematics). --- Two-dimensional space. --- Vector bundle. --- Vector space. --- Weak topology.

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This book is concerned with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them. These areas are the theory of linear differential equations in one complex variable with polynomial coefficients, and the theory of one parameter families of exponential sums over finite fields. After reviewing some results from representation theory, the book discusses results about differential equations and their differential galois groups (G) and one-parameter families of exponential sums and their geometric monodromy groups (G). The final part of the book is devoted to comparison theorems relating G and G of suitably "corresponding" situations, which provide a systematic explanation of the remarkable "coincidences" found "by hand" in the hypergeometric case.

Exponential sums. --- Differential equations. --- Adjoint representation. --- Algebraic geometry. --- Algebraic integer. --- Algebraically closed field. --- Automorphism. --- Base change. --- Bernard Dwork. --- Big O notation. --- Bijection. --- Calculation. --- Characteristic polynomial. --- Codimension. --- Coefficient. --- Cohomology. --- Comparison theorem. --- Complex manifold. --- Conjugacy class. --- Connected component (graph theory). --- Convolution. --- Determinant. --- Diagram (category theory). --- Differential Galois theory. --- Differential equation. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Divisor. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Equation. --- Euler characteristic. --- Existential quantification. --- Exponential sum. --- Fiber bundle. --- Field of fractions. --- Finite field. --- Formal power series. --- Fourier transform. --- Fundamental group. --- Fundamental representation. --- Galois extension. --- Galois group. --- Gauss sum. --- Generic point. --- Group theory. --- Homomorphism. --- Hypergeometric function. --- Identity component. --- Identity element. --- Integer. --- Irreducibility (mathematics). --- Irreducible representation. --- Isogeny. --- Isomorphism class. --- L-function. --- Laurent polynomial. --- Lie algebra. --- Logarithm. --- Mathematical induction. --- Matrix coefficient. --- Maximal compact subgroup. --- Maximal torus. --- Mellin transform. --- Monic polynomial. --- Monodromy theorem. --- Monodromy. --- Monomial. --- Natural number. --- Normal subgroup. --- P-adic number. --- Permutation. --- Polynomial. --- Prime number. --- Pullback. --- Quotient group. --- Reductive group. --- Regular singular point. --- Representation theory. --- Ring homomorphism. --- Root of unity. --- Scientific notation. --- Set (mathematics). --- Sheaf (mathematics). --- Special case. --- Subcategory. --- Subgroup. --- Subring. --- Subset. --- Summation. --- Surjective function. --- Symmetric group. --- Tensor product. --- Theorem. --- Theory. --- Three-dimensional space (mathematics). --- Torsor (algebraic geometry). --- Trichotomy (mathematics). --- Unitarian trick. --- Unitary group. --- Variable (mathematics).

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Recent developments in diverse areas of mathematics suggest the study of a certain class of extensions of C*-algebras. Here, Ronald Douglas uses methods from homological algebra to study this collection of extensions. He first shows that equivalence classes of the extensions of the compact metrizable space X form an abelian group Ext (X). Second, he shows that the correspondence X ⃗ Ext (X) defines a homotopy invariant covariant functor which can then be used to define a generalized homology theory. Establishing the periodicity of order two, the author shows, following Atiyah, that a concrete realization of K-homology is obtained.

Analytical spaces --- 517.986 --- Topological algebras. Theory of infinite-dimensional representations --- Algebra, Homological. --- C*-algebras. --- K-theory. --- 517.986 Topological algebras. Theory of infinite-dimensional representations --- Algebra, Homological --- C*-algebras --- K-theory --- Algebraic topology --- Homology theory --- Algebras, C star --- Algebras, W star --- C star algebras --- W star algebras --- W*-algebras --- Banach algebras --- Homological algebra --- Algebra, Abstract --- K-théorie. --- Homologie. --- Addition. --- Affine transformation. --- Algebraic topology. --- Atiyah–Singer index theorem. --- Automorphism. --- Banach algebra. --- Bijection. --- Boundary value problem. --- Bundle map. --- C*-algebra. --- Calculation. --- Cardinal number. --- Category of abelian groups. --- Characteristic class. --- Chern class. --- Clifford algebra. --- Coefficient. --- Cohomology. --- Compact operator. --- Completely positive map. --- Contact geometry. --- Continuous function. --- Corollary. --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Differential operator. --- Dimension (vector space). --- Dimension function. --- Dimension. --- Direct integral. --- Direct proof. --- Eigenvalues and eigenvectors. --- Equivalence class. --- Equivalence relation. --- Essential spectrum. --- Euler class. --- Exact sequence. --- Existential quantification. --- Fiber bundle. --- Finite group. --- Fredholm operator. --- Fredholm. --- Free abelian group. --- Fundamental class. --- Fundamental group. --- Hardy space. --- Hermann Weyl. --- Hilbert space. --- Homological algebra. --- Homology (mathematics). --- Homomorphism. --- Homotopy. --- Ideal (ring theory). --- Inner automorphism. --- Irreducible representation. --- K-group. --- Lebesgue space. --- Locally compact group. --- Maximal compact subgroup. --- Michael Atiyah. --- Monomorphism. --- Morphism. --- Natural number. --- Natural transformation. --- Normal operator. --- Operator algebra. --- Operator norm. --- Operator theory. --- Orthogonal group. --- Pairing. --- Piecewise linear manifold. --- Polynomial. --- Pontryagin class. --- Positive and negative parts. --- Positive map. --- Pseudo-differential operator. --- Quaternion. --- Quotient algebra. --- Self-adjoint operator. --- Self-adjoint. --- Simply connected space. --- Smooth structure. --- Special case. --- Stein manifold. --- Strong topology. --- Subalgebra. --- Subgroup. --- Subset. --- Summation. --- Tangent bundle. --- Theorem. --- Todd class. --- Topology. --- Torsion subgroup. --- Unitary operator. --- Universal coefficient theorem. --- Variable (mathematics). --- Von Neumann algebra. --- Homology theory. --- Homologie --- K-théorie --- C etoile-algebres

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This book is an expanded version of the Hermann Weyl Lectures given at the Institute for Advanced Study in January 1986. It outlines some of what is now known about irreducible unitary representations of real reductive groups, providing fairly complete definitions and references, and sketches (at least) of most proofs. The first half of the book is devoted to the three more or less understood constructions of such representations: parabolic induction, complementary series, and cohomological parabolic induction. This culminates in the description of all irreducible unitary representation of the general linear groups. For other groups, one expects to need a new construction, giving "unipotent representations." The latter half of the book explains the evidence for that expectation and suggests a partial definition of unipotent representations.

Lie groups --- Representations of Lie groups --- Lie groups. --- Representations of Lie groups. --- 512.81 --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- 512.81 Lie groups --- Abelian group. --- Adjoint representation. --- Annihilator (ring theory). --- Atiyah–Singer index theorem. --- Automorphic form. --- Automorphism. --- Cartan subgroup. --- Circle group. --- Class function (algebra). --- Classification theorem. --- Cohomology. --- Commutator subgroup. --- Complete metric space. --- Complex manifold. --- Conjugacy class. --- Cotangent space. --- Dimension (vector space). --- Discrete series representation. --- Dixmier conjecture. --- Dolbeault cohomology. --- Duality (mathematics). --- Eigenvalues and eigenvectors. --- Exponential map (Lie theory). --- Exponential map (Riemannian geometry). --- Exterior algebra. --- Function space. --- Group homomorphism. --- Harmonic analysis. --- Hecke algebra. --- Hilbert space. --- Hodge theory. --- Holomorphic function. --- Holomorphic vector bundle. --- Homogeneous space. --- Homomorphism. --- Induced representation. --- Infinitesimal character. --- Inner automorphism. --- Invariant subspace. --- Irreducibility (mathematics). --- Irreducible representation. --- Isometry group. --- Isometry. --- K-finite. --- Kazhdan–Lusztig polynomial. --- Langlands decomposition. --- Lie algebra cohomology. --- Lie algebra representation. --- Lie algebra. --- Lie group action. --- Lie group. --- Mathematical induction. --- Maximal compact subgroup. --- Measure (mathematics). --- Minkowski space. --- Nilpotent group. --- Orbit method. --- Orthogonal group. --- Parabolic induction. --- Principal homogeneous space. --- Principal series representation. --- Projective space. --- Pseudo-Riemannian manifold. --- Pullback (category theory). --- Ramanujan–Petersson conjecture. --- Reductive group. --- Regularity theorem. --- Representation of a Lie group. --- Representation theorem. --- Representation theory. --- Riemann sphere. --- Riemannian manifold. --- Schwartz space. --- Semisimple Lie algebra. --- Sheaf (mathematics). --- Sign (mathematics). --- Special case. --- Spectral theory. --- Sub"ient. --- Subgroup. --- Support (mathematics). --- Symplectic geometry. --- Symplectic group. --- Symplectic vector space. --- Tangent space. --- Tautological bundle. --- Theorem. --- Topological group. --- Topological space. --- Trivial representation. --- Unitary group. --- Unitary matrix. --- Unitary representation. --- Universal enveloping algebra. --- Vector bundle. --- Weyl algebra. --- Weyl character formula. --- Weyl group. --- Zariski's main theorem. --- Zonal spherical function. --- Représentations de groupes de Lie --- Groupes de lie --- Representation des groupes de lie

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The results established in this book constitute a new departure in ergodic theory and a significant expansion of its scope. Traditional ergodic theorems focused on amenable groups, and relied on the existence of an asymptotically invariant sequence in the group, the resulting maximal inequalities based on covering arguments, and the transference principle. Here, Alexander Gorodnik and Amos Nevo develop a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. Simple general conditions on the spectral theory of the group and the regularity of the averaging sets are formulated, which suffice to guarantee convergence to the ergodic mean. In particular, this approach gives a complete solution to the problem of establishing mean and pointwise ergodic theorems for the natural averages on semisimple algebraic groups and on their discrete lattice subgroups. Furthermore, an explicit quantitative rate of convergence to the ergodic mean is established in many cases. The topic of this volume lies at the intersection of several mathematical fields of fundamental importance. These include ergodic theory and dynamics of non-amenable groups, harmonic analysis on semisimple algebraic groups and their homogeneous spaces, quantitative non-Euclidean lattice point counting problems and their application to number theory, as well as equidistribution and non-commutative Diophantine approximation. Many examples and applications are provided in the text, demonstrating the usefulness of the results established.

Dynamics. --- Ergodic theory. --- Harmonic analysis. --- Lattice theory. --- Lie groups. --- Ergodic theory --- Lie groups --- Lattice theory --- Harmonic analysis --- Dynamics --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Dynamical systems --- Kinetics --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Lattices (Mathematics) --- Space lattice (Mathematics) --- Structural analysis (Mathematics) --- Groups, Lie --- Ergodic transformations --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Banach algebras --- Mathematical analysis --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Algebra, Abstract --- Algebra, Boolean --- Group theory --- Set theory --- Topology --- Transformations (Mathematics) --- Crystallography, Mathematical --- Lie algebras --- Symmetric spaces --- Topological groups --- Continuous groups --- Mathematical physics --- Measure theory --- Absolute continuity. --- Algebraic group. --- Amenable group. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Automorphism. --- Borel set. --- Bounded function. --- Bounded operator. --- Bounded set (topological vector space). --- Congruence subgroup. --- Continuous function. --- Convergence of random variables. --- Convolution. --- Coset. --- Counting problem (complexity). --- Counting. --- Differentiable function. --- Dimension (vector space). --- Diophantine approximation. --- Direct integral. --- Direct product. --- Discrete group. --- Embedding. --- Equidistribution theorem. --- Ergodicity. --- Estimation. --- Explicit formulae (L-function). --- Family of sets. --- Haar measure. --- Hilbert space. --- Hyperbolic space. --- Induced representation. --- Infimum and supremum. --- Initial condition. --- Interpolation theorem. --- Invariance principle (linguistics). --- Invariant measure. --- Irreducible representation. --- Isometry group. --- Iwasawa group. --- Lattice (group). --- Lie algebra. --- Linear algebraic group. --- Linear space (geometry). --- Lipschitz continuity. --- Mass distribution. --- Mathematical induction. --- Maximal compact subgroup. --- Maximal ergodic theorem. --- Measure (mathematics). --- Mellin transform. --- Metric space. --- Monotonic function. --- Neighbourhood (mathematics). --- Normal subgroup. --- Number theory. --- One-parameter group. --- Operator norm. --- Orthogonal complement. --- P-adic number. --- Parametrization. --- Parity (mathematics). --- Pointwise convergence. --- Pointwise. --- Principal homogeneous space. --- Principal series representation. --- Probability measure. --- Probability space. --- Probability. --- Rate of convergence. --- Regular representation. --- Representation theory. --- Resolution of singularities. --- Sobolev space. --- Special case. --- Spectral gap. --- Spectral method. --- Spectral theory. --- Square (algebra). --- Subgroup. --- Subsequence. --- Subset. --- Symmetric space. --- Tensor algebra. --- Tensor product. --- Theorem. --- Transfer principle. --- Unit sphere. --- Unit vector. --- Unitary group. --- Unitary representation. --- Upper and lower bounds. --- Variable (mathematics). --- Vector group. --- Vector space. --- Volume form. --- Word metric.