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Written and revised by D. B. A. Epstein.

Category theory. Homological algebra --- 515.14 --- Algebraic topology --- Homology theory. --- 515.14 Algebraic topology --- Cohomology theory --- Contrahomology theory --- Algebra homomorphism. --- Algebra over a field. --- Algebraic structure. --- Approximation. --- Axiom. --- Basis (linear algebra). --- CW complex. --- Cartesian product. --- Classical group. --- Coefficient. --- Cohomology operation. --- Cohomology ring. --- Cohomology. --- Commutative property. --- Complex number. --- Computation. --- Continuous function. --- Cup product. --- Cyclic group. --- Diagram (category theory). --- Dimension. --- Direct limit. --- Embedding. --- Existence theorem. --- Fibration. --- Homomorphism. --- Hopf algebra. --- Hopf invariant. --- Ideal (ring theory). --- Integer. --- Inverse limit. --- Manifold. --- Mathematics. --- Monomial. --- N-skeleton. --- Natural transformation. --- Permutation. --- Quaternion. --- Ring (mathematics). --- Scalar (physics). --- Special unitary group. --- Steenrod algebra. --- Stiefel manifold. --- Subgroup. --- Subset. --- Summation. --- Symmetric group. --- Symplectic group. --- Theorem. --- Uniqueness theorem. --- Upper and lower bounds. --- Vector field. --- Vector space. --- W0.

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One of the most exciting new subjects in Algebraic Number Theory and Arithmetic Algebraic Geometry is the theory of Euler systems. Euler systems are special collections of cohomology classes attached to p-adic Galois representations. Introduced by Victor Kolyvagin in the late 1980's in order to bound Selmer groups attached to p-adic representations, Euler systems have since been used to solve several key problems. These include certain cases of the Birch and Swinnerton-Dyer Conjecture and the Main Conjecture of Iwasawa Theory. Because Selmer groups play a central role in Arithmetic Algebraic Geometry, Euler systems should be a powerful tool in the future development of the field. Here, in the first book to appear on the subject, Karl Rubin presents a self-contained development of the theory of Euler systems. Rubin first reviews and develops the necessary facts from Galois cohomology. He then introduces Euler systems, states the main theorems, and develops examples and applications. The remainder of the book is devoted to the proofs of the main theorems as well as some further speculations. The book assumes a solid background in algebraic Number Theory, and is suitable as an advanced graduate text. As a research monograph it will also prove useful to number theorists and researchers in Arithmetic Algebraic Geometry.

Algebraic number theory. --- p-adic numbers. --- Numbers, p-adic --- Number theory --- p-adic analysis --- Galois cohomology --- Cohomologie galoisienne. --- Algebraic number theory --- p-adic numbers --- Abelian extension. --- Abelian variety. --- Absolute Galois group. --- Algebraic closure. --- Barry Mazur. --- Big O notation. --- Birch and Swinnerton-Dyer conjecture. --- Cardinality. --- Class field theory. --- Coefficient. --- Cohomology. --- Complex multiplication. --- Conjecture. --- Corollary. --- Cyclotomic field. --- Dimension (vector space). --- Divisibility rule. --- Eigenvalues and eigenvectors. --- Elliptic curve. --- Error term. --- Euler product. --- Euler system. --- Exact sequence. --- Existential quantification. --- Field of fractions. --- Finite set. --- Functional equation. --- Galois cohomology. --- Galois group. --- Galois module. --- Gauss sum. --- Global field. --- Heegner point. --- Ideal class group. --- Integer. --- Inverse limit. --- Inverse system. --- Karl Rubin. --- Local field. --- Mathematical induction. --- Maximal ideal. --- Modular curve. --- Modular elliptic curve. --- Natural number. --- Orthogonality. --- P-adic number. --- Pairing. --- Principal ideal. --- R-factor (crystallography). --- Ralph Greenberg. --- Remainder. --- Residue field. --- Ring of integers. --- Scientific notation. --- Selmer group. --- Subgroup. --- Tate module. --- Taylor series. --- Tensor product. --- Theorem. --- Upper and lower bounds. --- Victor Kolyvagin. --- Courbes elliptiques --- Nombres, Théorie des

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Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

Algebraic geometry --- Topology --- Toposes --- Categories (Mathematics) --- Categories (Mathematics). --- Toposes. --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Category theory (Mathematics) --- Topoi (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Functor theory --- Adjoint functors. --- Associative property. --- Base change map. --- Base change. --- CW complex. --- Canonical map. --- Cartesian product. --- Category of sets. --- Category theory. --- Coequalizer. --- Cofinality. --- Coherence theorem. --- Cohomology. --- Cokernel. --- Commutative property. --- Continuous function (set theory). --- Contractible space. --- Coproduct. --- Corollary. --- Derived category. --- Diagonal functor. --- Diagram (category theory). --- Dimension theory (algebra). --- Dimension theory. --- Dimension. --- Enriched category. --- Epimorphism. --- Equivalence class. --- Equivalence relation. --- Existence theorem. --- Existential quantification. --- Factorization system. --- Functor category. --- Functor. --- Fundamental group. --- Grothendieck topology. --- Grothendieck universe. --- Group homomorphism. --- Groupoid. --- Heyting algebra. --- Higher Topos Theory. --- Higher category theory. --- Homotopy category. --- Homotopy colimit. --- Homotopy group. --- Homotopy. --- I0. --- Inclusion map. --- Inductive dimension. --- Initial and terminal objects. --- Inverse limit. --- Isomorphism class. --- Kan extension. --- Limit (category theory). --- Localization of a category. --- Maximal element. --- Metric space. --- Model category. --- Monoidal category. --- Monoidal functor. --- Monomorphism. --- Monotonic function. --- Morphism. --- Natural transformation. --- Nisnevich topology. --- Noetherian topological space. --- Noetherian. --- O-minimal theory. --- Open set. --- Power series. --- Presheaf (category theory). --- Prime number. --- Pullback (category theory). --- Pushout (category theory). --- Quillen adjunction. --- Quotient by an equivalence relation. --- Regular cardinal. --- Retract. --- Right inverse. --- Sheaf (mathematics). --- Sheaf cohomology. --- Simplicial category. --- Simplicial set. --- Special case. --- Subcategory. --- Subset. --- Surjective function. --- Tensor product. --- Theorem. --- Topological space. --- Topology. --- Topos. --- Total order. --- Transitive relation. --- Universal property. --- Upper and lower bounds. --- Weak equivalence (homotopy theory). --- Yoneda lemma. --- Zariski topology. --- Zorn's lemma.

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This book describes the theory and applications of discrete orthogonal polynomials--polynomials that are orthogonal on a finite set. Unlike other books, Discrete Orthogonal Polynomials addresses completely general weight functions and presents a new methodology for handling the discrete weights case. J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller focus on asymptotic aspects of general, nonclassical discrete orthogonal polynomials and set out applications of current interest. Topics covered include the probability theory of discrete orthogonal polynomial ensembles and the continuum limit of the Toda lattice. The primary concern throughout is the asymptotic behavior of discrete orthogonal polynomials for general, nonclassical measures, in the joint limit where the degree increases as some fraction of the total number of points of collocation. The book formulates the orthogonality conditions defining these polynomials as a kind of Riemann-Hilbert problem and then generalizes the steepest descent method for such a problem to carry out the necessary asymptotic analysis.

Orthogonal polynomials --- Asymptotic theory --- Orthogonal polynomials -- Asymptotic theory. --- Polynomials. --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Asymptotic theory. --- Asymptotic theory of orthogonal polynomials --- Algebra --- Airy function. --- Analytic continuation. --- Analytic function. --- Ansatz. --- Approximation error. --- Approximation theory. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Asymptotic formula. --- Beta function. --- Boundary value problem. --- Calculation. --- Cauchy's integral formula. --- Cauchy–Riemann equations. --- Change of variables. --- Complex number. --- Complex plane. --- Correlation function. --- Degeneracy (mathematics). --- Determinant. --- Diagram (category theory). --- Discrete measure. --- Distribution function. --- Eigenvalues and eigenvectors. --- Equation. --- Estimation. --- Existential quantification. --- Explicit formulae (L-function). --- Factorization. --- Fredholm determinant. --- Functional derivative. --- Gamma function. --- Gradient descent. --- Harmonic analysis. --- Hermitian matrix. --- Homotopy. --- Hypergeometric function. --- I0. --- Identity matrix. --- Inequality (mathematics). --- Integrable system. --- Invariant measure. --- Inverse scattering transform. --- Invertible matrix. --- Jacobi matrix. --- Joint probability distribution. --- Lagrange multiplier. --- Lax equivalence theorem. --- Limit (mathematics). --- Linear programming. --- Lipschitz continuity. --- Matrix function. --- Maxima and minima. --- Monic polynomial. --- Monotonic function. --- Morera's theorem. --- Neumann series. --- Number line. --- Orthogonal polynomials. --- Orthogonality. --- Orthogonalization. --- Parameter. --- Parametrix. --- Pauli matrices. --- Pointwise convergence. --- Pointwise. --- Polynomial. --- Potential theory. --- Probability distribution. --- Probability measure. --- Probability theory. --- Probability. --- Proportionality (mathematics). --- Quantity. --- Random matrix. --- Random variable. --- Rate of convergence. --- Rectangle. --- Rhombus. --- Riemann surface. --- Special case. --- Spectral theory. --- Statistic. --- Subset. --- Theorem. --- Toda lattice. --- Trace (linear algebra). --- Trace class. --- Transition point. --- Triangular matrix. --- Trigonometric functions. --- Uniform continuity. --- Unit vector. --- Upper and lower bounds. --- Upper half-plane. --- Variational inequality. --- Weak solution. --- Weight function. --- Wishart distribution. --- Orthogonal polynomials - Asymptotic theory

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Using only the very elementary framework of finite probability spaces, this book treats a number of topics in the modern theory of stochastic processes. This is made possible by using a small amount of Abraham Robinson's nonstandard analysis and not attempting to convert the results into conventional form.

Martingales (Mathematics) --- Stochastic processes. --- Probabilities. --- Martingales (Mathematics). --- Stochastic processes --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Random processes --- Probabilities --- Abraham Robinson. --- Absolute value. --- Addition. --- Algebra of random variables. --- Almost surely. --- Axiom. --- Axiomatic system. --- Borel set. --- Bounded function. --- Cantor's diagonal argument. --- Cardinality. --- Cartesian product. --- Central limit theorem. --- Chebyshev's inequality. --- Compact space. --- Contradiction. --- Convergence of random variables. --- Corollary. --- Correlation coefficient. --- Counterexample. --- Dimension (vector space). --- Dimension. --- Division by zero. --- Elementary function. --- Estimation. --- Existential quantification. --- Family of sets. --- Finite set. --- Hyperplane. --- Idealization. --- Independence (probability theory). --- Indicator function. --- Infinitesimal. --- Internal set theory. --- Joint probability distribution. --- Law of large numbers. --- Linear function. --- Martingale (probability theory). --- Mathematical induction. --- Mathematician. --- Mathematics. --- Measure (mathematics). --- N0. --- Natural number. --- Non-standard analysis. --- Norm (mathematics). --- Orthogonal complement. --- Parameter. --- Path space. --- Predictable process. --- Probability distribution. --- Probability measure. --- Probability space. --- Probability theory. --- Probability. --- Product topology. --- Projection (linear algebra). --- Quadratic variation. --- Random variable. --- Real number. --- Requirement. --- Scientific notation. --- Sequence. --- Set (mathematics). --- Significant figures. --- Special case. --- Standard deviation. --- Statistical mechanics. --- Stochastic process. --- Subalgebra. --- Subset. --- Summation. --- Theorem. --- Theory. --- Total variation. --- Transfer principle. --- Transfinite number. --- Trigonometric functions. --- Upper and lower bounds. --- Variable (mathematics). --- Variance. --- Vector space. --- W0. --- Wiener process. --- Without loss of generality.