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This book offers a self-contained account of the 3-manifold invariants arising from the original Jones polynomial. These are the Witten-Reshetikhin-Turaev and the Turaev-Viro invariants. Starting from the Kauffman bracket model for the Jones polynomial and the diagrammatic Temperley-Lieb algebra, higher-order polynomial invariants of links are constructed and combined to form the 3-manifold invariants. The methods in this book are based on a recoupling theory for the Temperley-Lieb algebra. This recoupling theory is a q-deformation of the SU(2) spin networks of Roger Penrose. The recoupling theory is developed in a purely combinatorial and elementary manner. Calculations are based on a reformulation of the Kirillov-Reshetikhin shadow world, leading to expressions for all the invariants in terms of state summations on 2-cell complexes. Extensive tables of the invariants are included. Manifolds in these tables are recognized by surgery presentations and by means of 3-gems (graph encoded 3-manifolds) in an approach pioneered by Sostenes Lins. The appendices include information about gems, examples of distinct manifolds with the same invariants, and applications to the Turaev-Viro invariant and to the Crane-Yetter invariant of 4-manifolds.

Drie-menigvuldigheden (Topologie) --- Knopentheorie --- Knot theory --- Noeuds [Theorie des ] --- Three-manifolds (Topology) --- Trois-variétés (Topologie) --- Knot theory. --- Algebraic topology --- Invariants --- Mathematics --- Invariants (Mathematics) --- Invariants. --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Low-dimensional topology --- Topological manifolds --- Knots (Topology) --- 3-manifold. --- Addition. --- Algorithm. --- Ambient isotopy. --- Axiom. --- Backslash. --- Barycentric subdivision. --- Bijection. --- Bipartite graph. --- Borromean rings. --- Boundary parallel. --- Bracket polynomial. --- Calculation. --- Canonical form. --- Cartesian product. --- Cobordism. --- Coefficient. --- Combination. --- Commutator. --- Complex conjugate. --- Computation. --- Connected component (graph theory). --- Connected sum. --- Cubic graph. --- Diagram (category theory). --- Dimension. --- Disjoint sets. --- Disjoint union. --- Elaboration. --- Embedding. --- Equation. --- Equivalence class. --- Explicit formula. --- Explicit formulae (L-function). --- Factorial. --- Fundamental group. --- Graph (discrete mathematics). --- Graph embedding. --- Handlebody. --- Homeomorphism. --- Homology (mathematics). --- Identity element. --- Intersection form (4-manifold). --- Inverse function. --- Jones polynomial. --- Kirby calculus. --- Line segment. --- Linear independence. --- Matching (graph theory). --- Mathematical physics. --- Mathematical proof. --- Mathematics. --- Maxima and minima. --- Monograph. --- Natural number. --- Network theory. --- Notation. --- Numerical analysis. --- Orientability. --- Orthogonality. --- Pairing. --- Pairwise. --- Parametrization. --- Parity (mathematics). --- Partition function (mathematics). --- Permutation. --- Poincaré conjecture. --- Polyhedron. --- Quantum group. --- Quantum invariant. --- Recoupling. --- Recursion. --- Reidemeister move. --- Result. --- Roger Penrose. --- Root of unity. --- Scientific notation. --- Sequence. --- Significant figures. --- Simultaneous equations. --- Smoothing. --- Special case. --- Sphere. --- Spin network. --- Summation. --- Symmetric group. --- Tetrahedron. --- The Geometry Center. --- Theorem. --- Theory. --- Three-dimensional space (mathematics). --- Time complexity. --- Tubular neighborhood. --- Two-dimensional space. --- Vector field. --- Vector space. --- Vertex (graph theory). --- Winding number. --- Writhe.

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A survey, thorough and timely, of the singularities of two-dimensional normal complex analytic varieties, the volume summarizes the results obtained since Hirzebruch's thesis (1953) and presents new contributions. First, the singularity is resolved and shown to be classified by its resolution; then, resolutions are classed by the use of spaces with nilpotents; finally, the spaces with nilpotents are determined by means of the local ring structure of the singularity.

Algebraic geometry --- Analytic spaces --- SINGULARITIES (Mathematics) --- 512.76 --- Singularities (Mathematics) --- Geometry, Algebraic --- Spaces, Analytic --- Analytic functions --- Functions of several complex variables --- Birational geometry. Mappings etc. --- Analytic spaces. --- Singularities (Mathematics). --- 512.76 Birational geometry. Mappings etc. --- Birational geometry. Mappings etc --- Analytic function. --- Analytic set. --- Analytic space. --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Calculation. --- Chern class. --- Codimension. --- Coefficient. --- Cohomology. --- Compact Riemann surface. --- Complex manifold. --- Computation. --- Connected component (graph theory). --- Continuous function. --- Contradiction. --- Coordinate system. --- Corollary. --- Covering space. --- Dimension. --- Disjoint union. --- Divisor. --- Dual graph. --- Elliptic curve. --- Elliptic function. --- Embedding. --- Existential quantification. --- Factorization. --- Fiber bundle. --- Finite set. --- Formal power series. --- Hausdorff space. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Intersection (set theory). --- Intersection number (graph theory). --- Inverse limit. --- Irreducible component. --- Isolated singularity. --- Iteration. --- Lattice (group). --- Line bundle. --- Linear combination. --- Line–line intersection. --- Local coordinates. --- Local ring. --- Mathematical induction. --- Maximal ideal. --- Meromorphic function. --- Monic polynomial. --- Nilpotent. --- Normal bundle. --- Open set. --- Parameter. --- Plane curve. --- Pole (complex analysis). --- Power series. --- Presheaf (category theory). --- Projective line. --- Quadratic transformation. --- Quantity. --- Riemann surface. --- Riemann–Roch theorem. --- Several complex variables. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent space. --- Tensor algebra. --- Theorem. --- Topological space. --- Transition function. --- Two-dimensional space. --- Variable (mathematics). --- Zero divisor. --- Zero of a function. --- Zero set. --- Variétés complexes --- Espaces analytiques

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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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This book is a spectacular introduction to the modern mathematical discipline known as the Theory of Games. Harold Kuhn first presented these lectures at Princeton University in 1952. They succinctly convey the essence of the theory, in part through the prism of the most exciting developments at its frontiers half a century ago. Kuhn devotes considerable space to topics that, while not strictly the subject matter of game theory, are firmly bound to it. These are taken mainly from the geometry of convex sets and the theory of probability distributions. The book opens by addressing "matrix games," a name first introduced in these lectures as an abbreviation for two-person, zero-sum games in normal form with a finite number of pure strategies. It continues with a treatment of games in extensive form, using a model introduced by the author in 1950 that quickly supplanted von Neumann and Morgenstern's cumbersome approach. A final section deals with games that have an infinite number of pure strategies for the two players. Throughout, the theory is generously illustrated with examples, and exercises test the reader's understanding. A historical note caps off each chapter. For readers familiar with the calculus and with elementary matrix theory or vector analysis, this book offers an indispensable store of vital insights on a subject whose importance has only grown with the years.

Operational research. Game theory --- Game theory --- 519.83 --- Theory of games --- 519.83 Theory of games --- Game theory. --- Games, Theory of --- Mathematical models --- Mathematics --- Abstract algebra. --- Addition. --- Algorithm. --- Almost surely. --- Analytic geometry. --- Axiom. --- Basic solution (linear programming). --- Big O notation. --- Bijection. --- Binary relation. --- Boundary (topology). --- Bounded set (topological vector space). --- Branch point. --- Calculation. --- Cardinality of the continuum. --- Cardinality. --- Cartesian coordinate system. --- Characteristic function (probability theory). --- Combination. --- Computation. --- Connectivity (graph theory). --- Constructive proof. --- Convex combination. --- Convex function. --- Convex hull. --- Convex set. --- Coordinate system. --- David Gale. --- Diagram (category theory). --- Differential equation. --- Dimension (vector space). --- Dimensional analysis. --- Disjoint sets. --- Distribution function. --- Embedding. --- Empty set. --- Enumeration. --- Equation. --- Equilibrium point. --- Equivalence relation. --- Estimation. --- Euclidean space. --- Existential quantification. --- Expected loss. --- Extreme point. --- Formal scheme. --- Fundamental theorem. --- Galois theory. --- Geometry. --- Hyperplane. --- Inequality (mathematics). --- Infimum and supremum. --- Integer. --- Iterative method. --- Line segment. --- Linear equation. --- Linear inequality. --- Matching Pennies. --- Mathematical induction. --- Mathematical optimization. --- Mathematical theory. --- Mathematician. --- Mathematics. --- Matrix (mathematics). --- Measure (mathematics). --- Min-max theorem. --- Minimum distance. --- Mutual exclusivity. --- Prediction. --- Probability distribution. --- Probability interpretations. --- Probability measure. --- Probability theory. --- Probability. --- Proof by contradiction. --- Quantity. --- Rank (linear algebra). --- Rational number. --- Real number. --- Requirement. --- Scientific notation. --- Sign (mathematics). --- Solution set. --- Special case. --- Statistics. --- Strategist. --- Strategy (game theory). --- Subset. --- Theorem. --- Theory of Games and Economic Behavior. --- Theory. --- Three-dimensional space (mathematics). --- Total order. --- Two-dimensional space. --- Union (set theory). --- Unit interval. --- Unit square. --- Vector Analysis. --- Vector calculus. --- Vector space.

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The original goal that ultimately led to this volume was the construction of "motivic cohomology theory," whose existence was conjectured by A. Beilinson and S. Lichtenbaum. This is achieved in the book's fourth paper, using results of the other papers whose additional role is to contribute to our understanding of various properties of algebraic cycles. The material presented provides the foundations for the recent proof of the celebrated "Milnor Conjecture" by Vladimir Voevodsky. The theory of sheaves of relative cycles is developed in the first paper of this volume. The theory of presheaves with transfers and more specifically homotopy invariant presheaves with transfers is the main theme of the second paper. The Friedlander-Lawson moving lemma for families of algebraic cycles appears in the third paper in which a bivariant theory called bivariant cycle cohomology is constructed. The fifth and last paper in the volume gives a proof of the fact that bivariant cycle cohomology groups are canonically isomorphic (in appropriate cases) to Bloch's higher Chow groups, thereby providing a link between the authors' theory and Bloch's original approach to motivic (co-)homology.

Bundeltheorie --- Cohomology [Sheaf ] --- Faisceaux [Théorie des ] --- Sheaf cohomology --- Sheaf theory --- Sheaves (Algebraic topology) --- Sheaves [Theory of ] --- Théorie des faisceaux --- Algebraic cycles --- Homology theory --- Algebraic cycles. --- Homology theory. --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Cycles, Algebraic --- Geometry, Algebraic --- Abelian category. --- Abelian group. --- Addition. --- Additive category. --- Adjoint functors. --- Affine space. --- Affine variety. --- Alexander Grothendieck. --- Algebraic K-theory. --- Algebraic cycle. --- Algebraically closed field. --- Andrei Suslin. --- Associative property. --- Base change. --- Category of abelian groups. --- Chain complex. --- Chow group. --- Closed immersion. --- Codimension. --- Coefficient. --- Cohomology. --- Cokernel. --- Commutative property. --- Commutative ring. --- Compactification (mathematics). --- Comparison theorem. --- Computation. --- Connected component (graph theory). --- Connected space. --- Corollary. --- Diagram (category theory). --- Dimension. --- Discrete valuation ring. --- Disjoint union. --- Divisor. --- Embedding. --- Endomorphism. --- Epimorphism. --- Exact sequence. --- Existential quantification. --- Field of fractions. --- Functor. --- Generic point. --- Geometry. --- Grothendieck topology. --- Homeomorphism. --- Homogeneous coordinates. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy. --- Injective sheaf. --- Irreducible component. --- K-theory. --- Mathematical induction. --- Mayer–Vietoris sequence. --- Milnor K-theory. --- Monoid. --- Monoidal category. --- Monomorphism. --- Morphism of schemes. --- Morphism. --- Motivic cohomology. --- Natural transformation. --- Nisnevich topology. --- Noetherian. --- Open set. --- Pairing. --- Perfect field. --- Permutation. --- Picard group. --- Presheaf (category theory). --- Projective space. --- Projective variety. --- Proper morphism. --- Quasi-projective variety. --- Residue field. --- Resolution of singularities. --- Scientific notation. --- Sheaf (mathematics). --- Simplicial complex. --- Simplicial set. --- Singular homology. --- Smooth scheme. --- Spectral sequence. --- Subcategory. --- Subgroup. --- Summation. --- Support (mathematics). --- Tensor product. --- Theorem. --- Topology. --- Triangulated category. --- Type theory. --- Universal coefficient theorem. --- Variable (mathematics). --- Vector bundle. --- Vladimir Voevodsky. --- Zariski topology. --- Zariski's main theorem. --- 512.73 --- 512.73 Cohomology theory of algebraic varieties and schemes --- Cohomology theory of algebraic varieties and schemes