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Locally symmetric spaces are generalizations of spaces of constant curvature. In this book the author presents the proof of a remarkable phenomenon, which he calls "strong rigidity": this is a stronger form of the deformation rigidity that has been investigated by Selberg, Calabi-Vesentini, Weil, Borel, and Raghunathan.The proof combines the theory of semi-simple Lie groups, discrete subgroups, the geometry of E. Cartan's symmetric Riemannian spaces, elements of ergodic theory, and the fundamental theorem of projective geometry as applied to Tit's geometries. In his proof the author introduces two new notions having independent interest: one is "pseudo-isometries"; the other is a notion of a quasi-conformal mapping over the division algebra K (K equals real, complex, quaternion, or Cayley numbers). The author attempts to make the account accessible to readers with diverse backgrounds, and the book contains capsule descriptions of the various theories that enter the proof.

Differential geometry. Global analysis --- Riemannian manifolds --- Symmetric spaces --- Rigidity (Geometry) --- 512 --- Lie groups --- Geometric rigidity --- Rigidity theorem --- Discrete geometry --- Spaces, Symmetric --- Geometry, Differential --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Manifolds (Mathematics) --- Groups, Lie --- Lie algebras --- Topological groups --- Algebra --- Lie groups. --- Riemannian manifolds. --- Symmetric spaces. --- Rigidity (Geometry). --- 512 Algebra --- Addition. --- Adjoint representation. --- Affine space. --- Approximation. --- Automorphism. --- Axiom. --- Big O notation. --- Boundary value problem. --- Cohomology. --- Compact Riemann surface. --- Compact space. --- Conjecture. --- Constant curvature. --- Corollary. --- Counterexample. --- Covering group. --- Covering space. --- Curvature. --- Diameter. --- Diffeomorphism. --- Differentiable function. --- Dimension. --- Direct product. --- Division algebra. --- Ergodicity. --- Erlangen program. --- Existence theorem. --- Exponential function. --- Finitely generated group. --- Fundamental domain. --- Fundamental group. --- Geometry. --- Half-space (geometry). --- Hausdorff distance. --- Hermitian matrix. --- Homeomorphism. --- Homomorphism. --- Hyperplane. --- Identity matrix. --- Inner automorphism. --- Isometry group. --- Jordan algebra. --- Matrix multiplication. --- Metric space. --- Morphism. --- Möbius transformation. --- Normal subgroup. --- Normalizing constant. --- Partially ordered set. --- Permutation. --- Projective space. --- Riemann surface. --- Riemannian geometry. --- Sectional curvature. --- Self-adjoint. --- Set function. --- Smoothness. --- Stereographic projection. --- Subgroup. --- Subset. --- Summation. --- Symmetric space. --- Tangent space. --- Tangent vector. --- Theorem. --- Topology. --- Tubular neighborhood. --- Two-dimensional space. --- Unit sphere. --- Vector group. --- Weyl group. --- Riemann, Variétés de --- Lie, Groupes de --- Geometrie differentielle globale --- Varietes riemanniennes

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The description for this book, Linear Inequalities and Related Systems. (AM-38), Volume 38, will be forthcoming.

Operational research. Game theory --- Linear programming. --- Matrices. --- Game theory. --- Games, Theory of --- Theory of games --- Mathematical models --- Mathematics --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Production scheduling --- Programming (Mathematics) --- Banach space. --- Basic solution (linear programming). --- Big O notation. --- Bilinear form. --- Boundary (topology). --- Brouwer fixed-point theorem. --- Characterization (mathematics). --- Coefficient. --- Combination. --- Computation. --- Computational problem. --- Convex combination. --- Convex cone. --- Convex hull. --- Convex set. --- Corollary. --- Correlation and dependence. --- Cramer's rule. --- Cyclic permutation. --- Dedekind cut. --- Degeneracy (mathematics). --- Determinant. --- Diagram (category theory). --- Dilworth's theorem. --- Dimension (vector space). --- Directional derivative. --- Disjoint sets. --- Doubly stochastic matrix. --- Dual space. --- Duality (mathematics). --- Duality (optimization). --- Eigenvalues and eigenvectors. --- Elementary proof. --- Equation solving. --- Equation. --- Equivalence class. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- Extreme point. --- Fixed-point theorem. --- Functional analysis. --- Fundamental theorem. --- General equilibrium theory. --- Hall's theorem. --- Hilbert space. --- Incidence matrix. --- Inequality (mathematics). --- Infimum and supremum. --- Invertible matrix. --- Kakutani fixed-point theorem. --- Lagrange multiplier. --- Linear equation. --- Linear inequality. --- Linear map. --- Linear space (geometry). --- Linear subspace. --- Loss function. --- Main diagonal. --- Mathematical induction. --- Mathematical optimization. --- Mathematical problem. --- Max-flow min-cut theorem. --- Maxima and minima. --- Maximal set. --- Maximum flow problem. --- Menger's theorem. --- Minor (linear algebra). --- Monotonic function. --- N-vector. --- Nonlinear programming. --- Nonnegative matrix. --- Parity (mathematics). --- Partially ordered set. --- Permutation matrix. --- Permutation. --- Polyhedron. --- Quantity. --- Representation theorem. --- Row and column vectors. --- Scientific notation. --- Sensitivity analysis. --- Set notation. --- Sign (mathematics). --- Simplex algorithm. --- Simultaneous equations. --- Solution set. --- Special case. --- Subset. --- Summation. --- System of linear equations. --- Theorem. --- Transpose. --- Unit sphere. --- Unit vector. --- Upper and lower bounds. --- Variable (mathematics). --- Vector space. --- Von Neumann's theorem.

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Ramsey theory is a fast-growing area of combinatorics with deep connections to other fields of mathematics such as topological dynamics, ergodic theory, mathematical logic, and algebra. The area of Ramsey theory dealing with Ramsey-type phenomena in higher dimensions is particularly useful. Introduction to Ramsey Spaces presents in a systematic way a method for building higher-dimensional Ramsey spaces from basic one-dimensional principles. It is the first book-length treatment of this area of Ramsey theory, and emphasizes applications for related and surrounding fields of mathematics, such as set theory, combinatorics, real and functional analysis, and topology. In order to facilitate accessibility, the book gives the method in its axiomatic form with examples that cover many important parts of Ramsey theory both finite and infinite. An exciting new direction for combinatorics, this book will interest graduate students and researchers working in mathematical subdisciplines requiring the mastery and practice of high-dimensional Ramsey theory.

Algebraic spaces. --- Ramsey theory. --- Ramsey theory --- Algebraic spaces --- Mathematics --- Algebra --- Physical Sciences & Mathematics --- Spaces, Algebraic --- Geometry, Algebraic --- Combinatorial analysis --- Graph theory --- Analytic set. --- Axiom of choice. --- Baire category theorem. --- Baire space. --- Banach space. --- Bijection. --- Binary relation. --- Boolean prime ideal theorem. --- Borel equivalence relation. --- Borel measure. --- Borel set. --- C0. --- Cantor cube. --- Cantor set. --- Cantor space. --- Cardinality. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Combinatorics. --- Compact space. --- Compactification (mathematics). --- Complete metric space. --- Completely metrizable space. --- Constructible universe. --- Continuous function (set theory). --- Continuous function. --- Corollary. --- Countable set. --- Counterexample. --- Decision problem. --- Dense set. --- Diagonalization. --- Dimension (vector space). --- Dimension. --- Discrete space. --- Disjoint sets. --- Dual space. --- Embedding. --- Equation. --- Equivalence relation. --- Existential quantification. --- Family of sets. --- Forcing (mathematics). --- Forcing (recursion theory). --- Gap theorem. --- Geometry. --- Ideal (ring theory). --- Infinite product. --- Lebesgue measure. --- Limit point. --- Lipschitz continuity. --- Mathematical induction. --- Mathematical problem. --- Mathematics. --- Metric space. --- Metrization theorem. --- Monotonic function. --- Natural number. --- Natural topology. --- Neighbourhood (mathematics). --- Null set. --- Open set. --- Order type. --- Partial function. --- Partially ordered set. --- Peano axioms. --- Point at infinity. --- Pointwise. --- Polish space. --- Probability measure. --- Product measure. --- Product topology. --- Property of Baire. --- Ramsey's theorem. --- Right inverse. --- Scalar multiplication. --- Schauder basis. --- Semigroup. --- Sequence. --- Sequential space. --- Set (mathematics). --- Set theory. --- Sperner family. --- Subsequence. --- Subset. --- Subspace topology. --- Support function. --- Symmetric difference. --- Theorem. --- Topological dynamics. --- Topological group. --- Topological space. --- Topology. --- Tree (data structure). --- Unit interval. --- Unit sphere. --- Variable (mathematics). --- Well-order. --- Zorn's lemma.

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The study of exponential sums over finite fields, begun by Gauss nearly two centuries ago, has been completely transformed in recent years by advances in algebraic geometry, culminating in Deligne's work on the Weil Conjectures. It now appears as a very attractive mixture of algebraic geometry, representation theory, and the sheaf-theoretic incarnations of such standard constructions of classical analysis as convolution and Fourier transform. The book is simultaneously an account of some of these ideas, techniques, and results, and an account of their application to concrete equidistribution questions concerning Kloosterman sums and Gauss sums.

Group theory --- Algebraic geometry --- Number theory --- 511.33 --- Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- 511.33 Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- Gaussian sums --- Homology theory --- Kloosterman sums --- Monodromy groups --- Kloostermann sums --- Sums, Kloosterman --- Sums, Kloostermann --- Exponential sums --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Gauss sums --- Sums, Gaussian --- Analytical and multiplicative number theory. Asymptotics. Sieves etc --- Gaussian sums. --- Kloosterman sums. --- Homology theory. --- Monodromy groups. --- Number theory. --- Nombres, Théorie des. --- Exponential sums. --- Sommes exponentielles. --- Arithmetic --- Arithmétique --- Geometry, Algebraic. --- Géométrie algébrique --- Abelian category. --- Absolute Galois group. --- Absolute value. --- Additive group. --- Adjoint representation. --- Affine variety. --- Algebraic group. --- Automorphic form. --- Automorphism. --- Big O notation. --- Cartan subalgebra. --- Characteristic polynomial. --- Classification theorem. --- Coefficient. --- Cohomology. --- Cokernel. --- Combination. --- Commutator. --- Compactification (mathematics). --- Complex Lie group. --- Complex number. --- Conjugacy class. --- Continuous function. --- Convolution theorem. --- Convolution. --- Determinant. --- Diagonal matrix. --- Dimension (vector space). --- Direct sum. --- Dual basis. --- Eigenvalues and eigenvectors. --- Empty set. --- Endomorphism. --- Equidistribution theorem. --- Estimation. --- Exactness. --- Existential quantification. --- Exponential sum. --- Exterior algebra. --- Faithful representation. --- Finite field. --- Finite group. --- Four-dimensional space. --- Frobenius endomorphism. --- Fundamental group. --- Fundamental representation. --- Galois group. --- Gauss sum. --- Homomorphism. --- Integer. --- Irreducibility (mathematics). --- Isomorphism class. --- Kloosterman sum. --- L-function. --- Leray spectral sequence. --- Lie algebra. --- Lie theory. --- Maximal compact subgroup. --- Method of moments (statistics). --- Monodromy theorem. --- Monodromy. --- Morphism. --- Multiplicative group. --- Natural number. --- Nilpotent. --- Open problem. --- P-group. --- Pairing. --- Parameter space. --- Parameter. --- Partially ordered set. --- Perfect field. --- Point at infinity. --- Polynomial ring. --- Prime number. --- Quotient group. --- Representation ring. --- Representation theory. --- Residue field. --- Riemann hypothesis. --- Root of unity. --- Sheaf (mathematics). --- Simple Lie group. --- Skew-symmetric matrix. --- Smooth morphism. --- Special case. --- Spin representation. --- Subgroup. --- Support (mathematics). --- Symmetric matrix. --- Symplectic group. --- Symplectic vector space. --- Tensor product. --- Theorem. --- Trace (linear algebra). --- Trivial representation. --- Variable (mathematics). --- Weil conjectures. --- Weyl character formula. --- Zariski topology.