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This book contains a valuable discussion of renormalization through the addition of counterterms to the Lagrangian, giving the first complete proof of the cancellation of all divergences in an arbitrary interaction. The author also introduces a new method of renormalizing an arbitrary Feynman amplitude, a method that is simpler than previous approaches and can be used to study the renormalized perturbation series in quantum field theory.

Mathematical physics. --- Quantum field theory. --- Addition. --- Adjoint. --- Amplitude. --- Analytic continuation. --- Analytic function. --- Antiparticle. --- C-number. --- Calculation. --- Change of variables. --- Classical electromagnetism. --- Coefficient. --- Commutative property. --- Compact space. --- Complex analysis. --- Complex number. --- Connectivity (graph theory). --- Constant term. --- Convolution. --- Derivative. --- Diagram (category theory). --- Differentiable function. --- Distribution (mathematics). --- Equation. --- Estimation. --- Explicit formulae (L-function). --- Fermion. --- Fock space. --- Formal power series. --- Fourier transform. --- Free field. --- Gauge theory. --- Graph theory. --- Hilbert space. --- Incidence matrix. --- Interaction picture. --- Invertible matrix. --- Irreducibility (mathematics). --- Isolated singularity. --- Lagrangian (field theory). --- Laurent series. --- Mathematical induction. --- Mathematics. --- Momentum. --- Monomial. --- Multiple integral. --- National Science Foundation. --- Notation. --- Parameter. --- Path integral formulation. --- Permutation. --- Polynomial. --- Power series. --- Probability. --- Propagator. --- Quadratic form. --- Quantity. --- Remainder. --- Renormalization. --- Requirement. --- S-matrix. --- Scattering amplitude. --- Scientific notation. --- Second quantization. --- Several complex variables. --- Simple extension. --- Special case. --- Subset. --- Subtraction. --- Suggestion. --- Summation. --- Taylor series. --- Tensor product. --- Theorem. --- Theory. --- Topological space. --- Translational symmetry. --- Tree (data structure). --- Uniform convergence. --- Vacuum expectation value. --- Vacuum state. --- Vacuum. --- Variable (mathematics). --- Vector field. --- Vector potential. --- Wick's theorem. --- Z0.

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This book gives a new foundation for the theory of links in 3-space modeled on the modern developmentby Jaco, Shalen, Johannson, Thurston et al. of the theory of 3-manifolds. The basic construction is a method of obtaining any link by "splicing" links of the simplest kinds, namely those whose exteriors are Seifert fibered or hyperbolic. This approach to link theory is particularly attractive since most invariants of links are additive under splicing.Specially distinguished from this viewpoint is the class of links, none of whose splice components is hyperbolic. It includes all links constructed by cabling and connected sums, in particular all links of singularities of complex plane curves. One of the main contributions of this monograph is the calculation of invariants of these classes of links, such as the Alexander polynomials, monodromy, and Seifert forms.

Algebraic geometry --- Differential geometry. Global analysis --- Link theory. --- Curves, Plane. --- SINGULARITIES (Mathematics) --- Curves, Plane --- Invariants --- Link theory --- Singularities (Mathematics) --- Geometry, Algebraic --- Low-dimensional topology --- Piecewise linear topology --- Higher plane curves --- Plane curves --- Invariants. --- 3-sphere. --- Alexander Grothendieck. --- Alexander polynomial. --- Algebraic curve. --- Algebraic equation. --- Algebraic geometry. --- Algebraic surface. --- Algorithm. --- Ambient space. --- Analytic function. --- Approximation. --- Big O notation. --- Call graph. --- Cartesian coordinate system. --- Characteristic polynomial. --- Closed-form expression. --- Cohomology. --- Computation. --- Conjecture. --- Connected sum. --- Contradiction. --- Coprime integers. --- Corollary. --- Curve. --- Cyclic group. --- Determinant. --- Diagram (category theory). --- Diffeomorphism. --- Dimension. --- Disjoint union. --- Eigenvalues and eigenvectors. --- Equation. --- Equivalence class. --- Euler number. --- Existential quantification. --- Exterior (topology). --- Fiber bundle. --- Fibration. --- Foliation. --- Fundamental group. --- Geometry. --- Graph (discrete mathematics). --- Ground field. --- Homeomorphism. --- Homology sphere. --- Identity matrix. --- Integer matrix. --- Intersection form (4-manifold). --- Isolated point. --- Isolated singularity. --- Jordan normal form. --- Knot theory. --- Mathematical induction. --- Monodromy matrix. --- Monodromy. --- N-sphere. --- Natural transformation. --- Newton polygon. --- Newton's method. --- Normal (geometry). --- Notation. --- Pairwise. --- Parametrization. --- Plane curve. --- Polynomial. --- Power series. --- Projective plane. --- Puiseux series. --- Quantity. --- Rational function. --- Resolution of singularities. --- Riemann sphere. --- Riemann surface. --- Root of unity. --- Scientific notation. --- Seifert surface. --- Set (mathematics). --- Sign (mathematics). --- Solid torus. --- Special case. --- Stereographic projection. --- Submanifold. --- Summation. --- Theorem. --- Three-dimensional space (mathematics). --- Topology. --- Torus knot. --- Torus. --- Tubular neighborhood. --- Unit circle. --- Unit vector. --- Unknot. --- Variable (mathematics).

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On Knots is a journey through the theory of knots, starting from the simplest combinatorial ideas--ideas arising from the representation of weaving patterns. From this beginning, topological invariants are constructed directly: first linking numbers, then the Conway polynomial and skein theory. This paves the way for later discussion of the recently discovered Jones and generalized polynomials. The central chapter, Chapter Six, is a miscellany of topics and recreations. Here the reader will find the quaternions and the belt trick, a devilish rope trick, Alhambra mosaics, Fibonacci trees, the topology of DNA, and the author's geometric interpretation of the generalized Jones Polynomial.Then come branched covering spaces, the Alexander polynomial, signature theorems, the work of Casson and Gordon on slice knots, and a chapter on knots and algebraic singularities.The book concludes with an appendix about generalized polynomials.

Knot theory --- Knots (Topology) --- Low-dimensional topology --- Knot theory. --- Algebraic topology --- 3-sphere. --- Addition theorem. --- Addition. --- Alexander polynomial. --- Algebraic variety. --- Algorithm. --- Ambient isotopy. --- Arf invariant. --- Basepoint. --- Bijection. --- Bilinear form. --- Borromean rings. --- Bracket polynomial. --- Braid group. --- Branched covering. --- Chiral knot. --- Chromatic polynomial. --- Cobordism. --- Codimension. --- Combination. --- Combinatorics. --- Complex analysis. --- Concentric. --- Conjecture. --- Connected sum. --- Conway polynomial (finite fields). --- Counting. --- Covering space. --- Cyclic group. --- Dense set. --- Determinant. --- Diagram (category theory). --- Diffeomorphism. --- Dimension. --- Disjoint union. --- Disk (mathematics). --- Dual graph. --- Elementary algebra. --- Embedding. --- Enumeration. --- Existential quantification. --- Exotic sphere. --- Fibration. --- Formal power series. --- Fundamental group. --- Geometric topology. --- Geometry and topology. --- Geometry. --- Group action. --- Homotopy. --- Integer. --- Intersection form (4-manifold). --- Isolated singularity. --- Jones polynomial. --- Knot complement. --- Knot group. --- Laws of Form. --- Lens space. --- Linking number. --- Manifold. --- Module (mathematics). --- Morwen Thistlethwaite. --- Normal bundle. --- Notation. --- Obstruction theory. --- Operator algebra. --- Pairing. --- Parity (mathematics). --- Partition function (mathematics). --- Planar graph. --- Point at infinity. --- Polynomial ring. --- Polynomial. --- Quantity. --- Rectangle. --- Reidemeister move. --- Remainder. --- Root of unity. --- Saddle point. --- Seifert surface. --- Singularity theory. --- Slice knot. --- Special case. --- Statistical mechanics. --- Substructure. --- Summation. --- Symmetry. --- Theorem. --- Three-dimensional space (mathematics). --- Topological space. --- Torus knot. --- Trefoil knot. --- Tubular neighborhood. --- Underpinning. --- Unknot. --- Variable (mathematics). --- Whitehead link. --- Wild knot. --- Writhe. --- Variétés topologiques --- Topologie combinatoire --- Theorie des noeuds

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The theory of Riemann surfaces has a geometric and an analytic part. The former deals with the axiomatic definition of a Riemann surface, methods of construction, topological equivalence, and conformal mappings of one Riemann surface on another. The analytic part is concerned with the existence and properties of functions that have a special character connected with the conformal structure, for instance: subharmonic, harmonic, and analytic functions.Originally published in 1960.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

515.16 --- 515.16 Topology of manifolds --- Topology of manifolds --- Riemann surfaces. --- Topology. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Surfaces, Riemann --- Functions --- Analytic function. --- Axiom of choice. --- Basis (linear algebra). --- Betti number. --- Big O notation. --- Bijection. --- Bilinear form. --- Bolzano–Weierstrass theorem. --- Boundary (topology). --- Boundary value problem. --- Bounded set (topological vector space). --- Branch point. --- Canonical basis. --- Cauchy sequence. --- Cauchy's integral formula. --- Characterization (mathematics). --- Coefficient. --- Commutator subgroup. --- Compact space. --- Compactification (mathematics). --- Conformal map. --- Connected space. --- Connectedness. --- Continuous function (set theory). --- Continuous function. --- Coset. --- Cross-cap. --- Dirichlet integral. --- Disjoint union. --- Elementary function. --- Elliptic surface. --- Exact differential. --- Existence theorem. --- Existential quantification. --- Extremal length. --- Family of sets. --- Finite intersection property. --- Finitely generated abelian group. --- Free group. --- Function (mathematics). --- Fundamental group. --- Green's function. --- Harmonic differential. --- Harmonic function. --- Harmonic measure. --- Heine–Borel theorem. --- Homeomorphism. --- Homology (mathematics). --- Ideal point. --- Infimum and supremum. --- Isolated point. --- Isolated singularity. --- Jordan curve theorem. --- Lebesgue integration. --- Limit point. --- Line segment. --- Linear independence. --- Linear map. --- Maximal set. --- Maximum principle. --- Meromorphic function. --- Metric space. --- Normal operator. --- Normal subgroup. --- Open set. --- Orientability. --- Orthogonal complement. --- Partition of unity. --- Point at infinity. --- Polyhedron. --- Positive harmonic function. --- Principal value. --- Projection (linear algebra). --- Projection (mathematics). --- Removable singularity. --- Riemann mapping theorem. --- Riemann surface. --- Semi-continuity. --- Sign (mathematics). --- Simplicial homology. --- Simply connected space. --- Singular homology. --- Skew-symmetric matrix. --- Special case. --- Subgroup. --- Subset. --- Summation. --- Support (mathematics). --- Taylor series. --- Theorem. --- Topological space. --- Triangle inequality. --- Uniform continuity. --- Uniformization theorem. --- Unit disk. --- Upper and lower bounds. --- Upper half-plane. --- Weyl's lemma (Laplace equation). --- Zorn's lemma.

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The description for this book, Seminar On Minimal Submanifolds. (AM-103), Volume 103, will be forthcoming.

Minimal submanifolds. --- A priori estimate. --- Analytic function. --- Banach space. --- Boundary (topology). --- Boundary value problem. --- Bounded set (topological vector space). --- Branch point. --- Cauchy–Riemann equations. --- Center manifold. --- Closed geodesic. --- Codimension. --- Coefficient. --- Cohomology. --- Compactness theorem. --- Comparison theorem. --- Configuration space. --- Conformal geometry. --- Conformal group. --- Conformal map. --- Continuous function. --- Cross product. --- Curve. --- Degeneracy (mathematics). --- Diffeomorphism. --- Differential form. --- Dirac operator. --- Discrete group. --- Divergence theorem. --- Eigenvalues and eigenvectors. --- Elementary proof. --- Equation. --- Existence theorem. --- Existential quantification. --- Exterior derivative. --- First variation. --- Free boundary problem. --- Fundamental group. --- Gauss map. --- Geodesic. --- Geometry. --- Group action. --- Hamiltonian mechanics. --- Harmonic function. --- Harmonic map. --- Hausdorff dimension. --- Hausdorff measure. --- Homotopy group. --- Homotopy. --- Hurewicz theorem. --- Hyperbolic 3-manifold. --- Hyperbolic manifold. --- Hyperbolic space. --- Hypersurface. --- Implicit function theorem. --- Infimum and supremum. --- Injective function. --- Inner automorphism. --- Isolated singularity. --- Isometry group. --- Isoperimetric problem. --- Klein bottle. --- Kleinian group. --- Limit set. --- Lipschitz continuity. --- Mapping class group. --- Maxima and minima. --- Maximum principle. --- Minimal surface of revolution. --- Minimal surface. --- Monotonic function. --- Möbius transformation. --- Norm (mathematics). --- Orthonormal basis. --- Parametric surface. --- Periodic function. --- Poincaré conjecture. --- Projection (linear algebra). --- Regularity theorem. --- Riemann surface. --- Riemannian manifold. --- Schwarz reflection principle. --- Second fundamental form. --- Semi-continuity. --- Simply connected space. --- Special case. --- Stein's lemma. --- Subalgebra. --- Subgroup. --- Submanifold. --- Subsequence. --- Support (mathematics). --- Symplectic manifold. --- Tangent space. --- Teichmüller space. --- Theorem. --- Trace (linear algebra). --- Uniformization. --- Uniqueness theorem. --- Variational principle. --- Yamabe problem.