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The description for this book, Topics in Transcendental Algebraic Geometry. (AM-106), Volume 106, will be forthcoming.

Geometry, Algebraic. --- Hodge theory. --- Torelli theorem. --- Géométrie algébrique --- Théorie de Hodge --- Geometry, Algebraic --- Hodge theory --- Torelli theorem --- 512.7 --- Torelli's theorem --- Curves, Algebraic --- Jacobians --- Complex manifolds --- Differentiable manifolds --- Homology theory --- Algebraic geometry --- Geometry --- Algebraic geometry. Commutative rings and algebras --- 512.7 Algebraic geometry. Commutative rings and algebras --- Géométrie algébrique --- Théorie de Hodge --- Abelian integral. --- Algebraic curve. --- Algebraic cycle. --- Algebraic equation. --- Algebraic geometry. --- Algebraic integer. --- Algebraic structure. --- Algebraic surface. --- Arithmetic genus. --- Arithmetic group. --- Asymptotic analysis. --- Automorphism. --- Base change. --- Bilinear form. --- Bilinear map. --- Cohomology. --- Combinatorics. --- Commutative diagram. --- Compactification (mathematics). --- Complete intersection. --- Complex manifold. --- Complex number. --- Computation. --- Deformation theory. --- Degeneracy (mathematics). --- Differentiable manifold. --- Dimension (vector space). --- Divisor (algebraic geometry). --- Divisor. --- Elliptic curve. --- Elliptic surface. --- Equation. --- Exact sequence. --- Fiber bundle. --- Function (mathematics). --- Fundamental class. --- Geometric genus. --- Geometry. --- Hermitian symmetric space. --- Hodge structure. --- Homology (mathematics). --- Homomorphism. --- Homotopy. --- Hypersurface. --- Intersection form (4-manifold). --- Intersection number. --- Irreducibility (mathematics). --- Isomorphism class. --- Jacobian variety. --- K3 surface. --- Kodaira dimension. --- Kronecker's theorem. --- Kummer surface. --- Kähler manifold. --- Lie algebra bundle. --- Lie algebra. --- Linear algebra. --- Linear algebraic group. --- Line–line intersection. --- Mathematical induction. --- Mathematical proof. --- Mathematics. --- Modular arithmetic. --- Module (mathematics). --- Moduli space. --- Monodromy matrix. --- Monodromy theorem. --- Monodromy. --- Nilpotent orbit. --- Normal function. --- Open set. --- Period mapping. --- Permutation group. --- Phillip Griffiths. --- Point at infinity. --- Pole (complex analysis). --- Polynomial. --- Projective space. --- Pullback (category theory). --- Quadric. --- Regular singular point. --- Resolution of singularities. --- Riemann–Roch theorem for surfaces. --- Scientific notation. --- Set (mathematics). --- Special case. --- Spectral sequence. --- Subgroup. --- Submanifold. --- Surface of general type. --- Surjective function. --- Tangent bundle. --- Theorem. --- Topology. --- Transcendental number. --- Vector space. --- Zariski topology. --- Zariski's main theorem.

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It is now some thirty years since Deligne first proved his general equidistribution theorem, thus establishing the fundamental result governing the statistical properties of suitably "pure" algebro-geometric families of character sums over finite fields (and of their associated L-functions). Roughly speaking, Deligne showed that any such family obeys a "generalized Sato-Tate law," and that figuring out which generalized Sato-Tate law applies to a given family amounts essentially to computing a certain complex semisimple (not necessarily connected) algebraic group, the "geometric monodromy group" attached to that family. Up to now, nearly all techniques for determining geometric monodromy groups have relied, at least in part, on local information. In Moments, Monodromy, and Perversity, Nicholas Katz develops new techniques, which are resolutely global in nature. They are based on two vital ingredients, neither of which existed at the time of Deligne's original work on the subject. The first is the theory of perverse sheaves, pioneered by Goresky and MacPherson in the topological setting and then brilliantly transposed to algebraic geometry by Beilinson, Bernstein, Deligne, and Gabber. The second is Larsen's Alternative, which very nearly characterizes classical groups by their fourth moments. These new techniques, which are of great interest in their own right, are first developed and then used to calculate the geometric monodromy groups attached to some quite specific universal families of (L-functions attached to) character sums over finite fields.

Monodromy groups. --- Sheaf theory. --- L-functions. --- Addition. --- Additive group. --- Affine space. --- Algebraic group. --- Algebraic integer. --- Algebraically closed field. --- Automorphism. --- Base change. --- Big O notation. --- Central moment. --- Change of base. --- Character sum. --- Classical group. --- Codimension. --- Computation. --- Conjecture. --- Conjugacy class. --- Constant function. --- Convolution. --- Corollary. --- Critical value. --- Dense set. --- Determinant. --- Dimension (vector space). --- Dimension. --- Diophantine equation. --- Direct sum. --- Discrete group. --- Disjoint sets. --- Divisor (algebraic geometry). --- Divisor. --- Eigenvalues and eigenvectors. --- Elliptic curve. --- Empty set. --- Equidistribution theorem. --- Existential quantification. --- Exponential sum. --- Faithful representation. --- Finite field. --- Finite group. --- Fourier transform. --- Function field. --- Function space. --- Generic point. --- Group theory. --- Hypersurface. --- Inequality (mathematics). --- Integer. --- Irreducible representation. --- Isomorphism class. --- L-function. --- Leray spectral sequence. --- Linear space (geometry). --- Linear subspace. --- Moment (mathematics). --- Monodromy. --- Morphism. --- Natural number. --- Normal subgroup. --- Orthogonal group. --- P-value. --- Parameter space. --- Parameter. --- Parity (mathematics). --- Partition of a set. --- Perverse sheaf. --- Polynomial. --- Power series. --- Prime number. --- Probability space. --- Probability theory. --- Proper morphism. --- Pullback (category theory). --- Random variable. --- Reductive group. --- Relative dimension. --- Root of unity. --- Scalar multiplication. --- Scientific notation. --- Set (mathematics). --- Sheaf (mathematics). --- Special case. --- Subgroup. --- Subobject. --- Subset. --- Summation. --- Surjective function. --- Symmetric group. --- Symplectic group. --- Tensor product. --- Theorem. --- Theory. --- Topology. --- Trace (linear algebra). --- Trivial group. --- Unipotent. --- Variable (mathematics). --- Variance. --- Vector space. --- Zariski 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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Algebraic K-theory describes a branch of algebra that centers about two functors. K0 and K1, which assign to each associative ring ∧ an abelian group K0∧ or K1∧ respectively. Professor Milnor sets out, in the present work, to define and study an analogous functor K2, also from associative rings to abelian groups. Just as functors K0 and K1 are important to geometric topologists, K2 is now considered to have similar topological applications. The exposition includes, besides K-theory, a considerable amount of related arithmetic.

Algebraic geometry --- Ordered algebraic structures --- Associative rings --- Abelian groups --- Functor theory --- Anneaux associatifs --- Groupes abéliens --- Foncteurs, Théorie des --- 512.73 --- 515.14 --- Functorial representation --- Algebra, Homological --- Categories (Mathematics) --- Functional analysis --- Transformations (Mathematics) --- Commutative groups --- Group theory --- Rings (Algebra) --- Cohomology theory of algebraic varieties and schemes --- Algebraic topology --- Abelian groups. --- Associative rings. --- Functor theory. --- 515.14 Algebraic topology --- 512.73 Cohomology theory of algebraic varieties and schemes --- Groupes abéliens --- Foncteurs, Théorie des --- Abelian group. --- Absolute value. --- Addition. --- Algebraic K-theory. --- Algebraic equation. --- Algebraic integer. --- Banach algebra. --- Basis (linear algebra). --- Big O notation. --- Circle group. --- Coefficient. --- Commutative property. --- Commutative ring. --- Commutator. --- Complex number. --- Computation. --- Congruence subgroup. --- Coprime integers. --- Cyclic group. --- Dedekind domain. --- Direct limit. --- Direct proof. --- Direct sum. --- Discrete valuation. --- Division algebra. --- Division ring. --- Elementary matrix. --- Elliptic function. --- Exact sequence. --- Existential quantification. --- Exterior algebra. --- Factorization. --- Finite group. --- Free abelian group. --- Function (mathematics). --- Fundamental group. --- Galois extension. --- Galois group. --- General linear group. --- Group extension. --- Hausdorff space. --- Homological algebra. --- Homomorphism. --- Homotopy. --- Ideal (ring theory). --- Ideal class group. --- Identity element. --- Identity matrix. --- Integral domain. --- Invertible matrix. --- Isomorphism class. --- K-theory. --- Kummer theory. --- Lattice (group). --- Left inverse. --- Local field. --- Local ring. --- Mathematics. --- Matsumoto's theorem. --- Maximal ideal. --- Meromorphic function. --- Monomial. --- Natural number. --- Noetherian. --- Normal subgroup. --- Number theory. --- Open set. --- Picard group. --- Polynomial. --- Prime element. --- Prime ideal. --- Projective module. --- Quadratic form. --- Quaternion. --- Quotient ring. --- Rational number. --- Real number. --- Right inverse. --- Ring of integers. --- Root of unity. --- Schur multiplier. --- Scientific notation. --- Simple algebra. --- Special case. --- Special linear group. --- Subgroup. --- Summation. --- Surjective function. --- Tensor product. --- Theorem. --- Topological K-theory. --- Topological group. --- Topological space. --- Topology. --- Torsion group. --- Variable (mathematics). --- Vector space. --- Wedderburn's theorem. --- Weierstrass function. --- Whitehead torsion. --- K-théorie