Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The description for this book, Calculus on Heisenberg Manifolds. (AM-119), Volume 119, will be forthcoming.

Hypoelliptic operators. --- Calculus. --- Differentiable manifolds. --- Adjoint. --- Affine transformation. --- Approximation. --- Asymptotic expansion. --- Calculation. --- Codimension. --- Complex geometry. --- Complex manifold. --- Computation. --- Convolution. --- De Rham cohomology. --- Derivative. --- Differentiable manifold. --- Differential operator. --- Dimension (vector space). --- Estimation. --- Fourier integral operator. --- Fourier transform. --- Function space. --- Heat equation. --- Heisenberg group. --- Hilbert space. --- Homogeneous function. --- Hypoelliptic operator. --- Identity element. --- Integration by parts. --- Invertible matrix. --- Manifold. --- Nilpotent group. --- Parametrix. --- Partial differential equation. --- Pointwise product. --- Pointwise. --- Polynomial. --- Principal part. --- Pseudo-differential operator. --- Riemannian manifold. --- Self-adjoint. --- Several complex variables. --- Singular integral. --- Smoothing. --- Structure constants. --- Subset. --- Summation. --- Tangent bundle. --- Theorem. --- Transpose. --- Unit circle. --- Vector field.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The intention of the authors is to examine the relationship between piecewise linear structure and differential structure: a relationship, they assert, that can be understood as a homotopy obstruction theory, and, hence, can be studied by using the traditional techniques of algebraic topology.Thus the book attacks the problem of existence and classification (up to isotopy) of differential structures compatible with a given combinatorial structure on a manifold. The problem is completely "solved" in the sense that it is reduced to standard problems of algebraic topology.The first part of the book is purely geometrical; it proves that every smoothing of the product of a manifold M and an interval is derived from an essentially unique smoothing of M. In the second part this result is used to translate the classification of smoothings into the problem of putting a linear structure on the tangent microbundle of M. This in turn is converted to the homotopy problem of classifying maps from M into a certain space PL/O. The set of equivalence classes of smoothings on M is given a natural abelian group structure.

Algebraic topology --- Piecewise linear topology --- Manifolds (Mathematics) --- Topologie linéaire par morceaux --- Variétés (Mathématiques) --- 515.16 --- PL topology --- Topology --- Geometry, Differential --- Topology of manifolds --- Piecewise linear topology. --- Manifolds (Mathematics). --- 515.16 Topology of manifolds --- Topologie linéaire par morceaux --- Variétés (Mathématiques) --- Affine transformation. --- Approximation. --- Associative property. --- Bijection. --- Bundle map. --- Classification theorem. --- Codimension. --- Coefficient. --- Cohomology. --- Commutative property. --- Computation. --- Convex cone. --- Convolution. --- Corollary. --- Counterexample. --- Diffeomorphism. --- Differentiable function. --- Differentiable manifold. --- Differential structure. --- Dimension. --- Direct proof. --- Division by zero. --- Embedding. --- Empty set. --- Equivalence class. --- Equivalence relation. --- Euclidean space. --- Existential quantification. --- Exponential map (Lie theory). --- Fiber bundle. --- Fibration. --- Functor. --- Grassmannian. --- H-space. --- Homeomorphism. --- Homotopy. --- Integral curve. --- Inverse problem. --- Isomorphism class. --- K0. --- Linearization. --- Manifold. --- Mathematical induction. --- Milnor conjecture. --- Natural transformation. --- Neighbourhood (mathematics). --- Normal bundle. --- Obstruction theory. --- Open set. --- Partition of unity. --- Piecewise linear. --- Polyhedron. --- Reflexive relation. --- Regular map (graph theory). --- Sheaf (mathematics). --- Smoothing. --- Smoothness. --- Special case. --- Submanifold. --- Tangent bundle. --- Tangent vector. --- Theorem. --- Topological manifold. --- Topological space. --- Topology. --- Transition function. --- Transitive relation. --- Vector bundle. --- Vector field. --- Variétés topologiques

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

In 1920, Pierre Fatou expressed the conjecture that--except for special cases--all critical points of a rational map of the Riemann sphere tend to periodic orbits under iteration. This conjecture remains the main open problem in the dynamics of iterated maps. For the logistic family x- ax(1-x), it can be interpreted to mean that for a dense set of parameters "a," an attracting periodic orbit exists. The same question appears naturally in science, where the logistic family is used to construct models in physics, ecology, and economics. In this book, Jacek Graczyk and Grzegorz Swiatek provide a rigorous proof of the Real Fatou Conjecture. In spite of the apparently elementary nature of the problem, its solution requires advanced tools of complex analysis. The authors have written a self-contained and complete version of the argument, accessible to someone with no knowledge of complex dynamics and only basic familiarity with interval maps. The book will thus be useful to specialists in real dynamics as well as to graduate students.

Geodesics (Mathematics) --- Polynomials. --- Mappings (Mathematics) --- Maps (Mathematics) --- Functions --- Functions, Continuous --- Topology --- Transformations (Mathematics) --- Algebra --- Geometry, Differential --- Global analysis (Mathematics) --- Mathematics --- Absolute value. --- Affine transformation. --- Algebraic function. --- Analytic continuation. --- Analytic function. --- Arithmetic. --- Automorphism. --- Big O notation. --- Bounded set (topological vector space). --- C0. --- Calculation. --- Canonical map. --- Change of variables. --- Chebyshev polynomials. --- Combinatorics. --- Commutative property. --- Complex number. --- Complex plane. --- Complex quadratic polynomial. --- Conformal map. --- Conjecture. --- Conjugacy class. --- Conjugate points. --- Connected component (graph theory). --- Connected space. --- Continuous function. --- Corollary. --- Covering space. --- Critical point (mathematics). --- Dense set. --- Derivative. --- Diffeomorphism. --- Dimension. --- Disjoint sets. --- Disjoint union. --- Disk (mathematics). --- Equicontinuity. --- Estimation. --- Existential quantification. --- Fibonacci. --- Functional equation. --- Fundamental domain. --- Generalization. --- Great-circle distance. --- Hausdorff distance. --- Holomorphic function. --- Homeomorphism. --- Homotopy. --- Hyperbolic function. --- Imaginary number. --- Implicit function theorem. --- Injective function. --- Integer. --- Intermediate value theorem. --- Interval (mathematics). --- Inverse function. --- Irreducible polynomial. --- Iteration. --- Jordan curve theorem. --- Julia set. --- Limit of a sequence. --- Linear map. --- Local diffeomorphism. --- Mathematical induction. --- Mathematical proof. --- Maxima and minima. --- Meromorphic function. --- Moduli (physics). --- Monomial. --- Monotonic function. --- Natural number. --- Neighbourhood (mathematics). --- Open set. --- Parameter. --- Periodic function. --- Periodic point. --- Phase space. --- Point at infinity. --- Polynomial. --- Projection (mathematics). --- Quadratic function. --- Quadratic. --- Quasiconformal mapping. --- Renormalization. --- Riemann sphere. --- Riemann surface. --- Schwarzian derivative. --- Scientific notation. --- Subsequence. --- Theorem. --- Theory. --- Topological conjugacy. --- Topological entropy. --- Topology. --- Union (set theory). --- Unit circle. --- Unit disk. --- Upper and lower bounds. --- Upper half-plane. --- Z0.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The recent introduction of the Seiberg-Witten invariants of smooth four-manifolds has revolutionized the study of those manifolds. The invariants are gauge-theoretic in nature and are close cousins of the much-studied SU(2)-invariants defined over fifteen years ago by Donaldson. On a practical level, the new invariants have proved to be more powerful and have led to a vast generalization of earlier results. This book is an introduction to the Seiberg-Witten invariants. The work begins with a review of the classical material on Spin c structures and their associated Dirac operators. Next comes a discussion of the Seiberg-Witten equations, which is set in the context of nonlinear elliptic operators on an appropriate infinite dimensional space of configurations. It is demonstrated that the space of solutions to these equations, called the Seiberg-Witten moduli space, is finite dimensional, and its dimension is then computed. In contrast to the SU(2)-case, the Seiberg-Witten moduli spaces are shown to be compact. The Seiberg-Witten invariant is then essentially the homology class in the space of configurations represented by the Seiberg-Witten moduli space. The last chapter gives a flavor for the applications of these new invariants by computing the invariants for most Kahler surfaces and then deriving some basic toological consequences for these surfaces.

Four-manifolds (Topology) --- Seiberg-Witten invariants. --- Mathematical physics. --- Physical mathematics --- Physics --- Invariants --- 4-dimensional manifolds (Topology) --- 4-manifolds (Topology) --- Four dimensional manifolds (Topology) --- Manifolds, Four dimensional --- Low-dimensional topology --- Topological manifolds --- Mathematics --- Affine space. --- Affine transformation. --- Algebra bundle. --- Algebraic surface. --- Almost complex manifold. --- Automorphism. --- Banach space. --- Clifford algebra. --- Cohomology. --- Cokernel. --- Complex dimension. --- Complex manifold. --- Complex plane. --- Complex projective space. --- Complex vector bundle. --- Complexification (Lie group). --- Computation. --- Configuration space. --- Conjugate transpose. --- Covariant derivative. --- Curvature form. --- Curvature. --- Differentiable manifold. --- Differential topology. --- Dimension (vector space). --- Dirac equation. --- Dirac operator. --- Division algebra. --- Donaldson theory. --- Duality (mathematics). --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Elliptic surface. --- Equation. --- Fiber bundle. --- Frenet–Serret formulas. --- Gauge fixing. --- Gauge theory. --- Gaussian curvature. --- Geometry. --- Group homomorphism. --- Hilbert space. --- Hodge index theorem. --- Homology (mathematics). --- Homotopy. --- Identity (mathematics). --- Implicit function theorem. --- Intersection form (4-manifold). --- Inverse function theorem. --- Isomorphism class. --- K3 surface. --- Kähler manifold. --- Levi-Civita connection. --- Lie algebra. --- Line bundle. --- Linear map. --- Linear space (geometry). --- Linearization. --- Manifold. --- Mathematical induction. --- Moduli space. --- Multiplication theorem. --- Neighbourhood (mathematics). --- One-form. --- Open set. --- Orientability. --- Orthonormal basis. --- Parameter space. --- Parametric equation. --- Parity (mathematics). --- Partial derivative. --- Principal bundle. --- Projection (linear algebra). --- Pullback (category theory). --- Quadratic form. --- Quaternion algebra. --- Quotient space (topology). --- Riemann surface. --- Riemannian manifold. --- Sard's theorem. --- Sign (mathematics). --- Sobolev space. --- Spin group. --- Spin representation. --- Spin structure. --- Spinor field. --- Subgroup. --- Submanifold. --- Surjective function. --- Symplectic geometry. --- Symplectic manifold. --- Tangent bundle. --- Tangent space. --- Tensor product. --- Theorem. --- Three-dimensional space (mathematics). --- Trace (linear algebra). --- Transversality (mathematics). --- Two-form. --- Zariski tangent space.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Charles Favre and Thomas Gauthier present new mathematical research in the field of arithmetic dynamics. Specifically, the authors study one-dimensional algebraic families of pairs given by a polynomial with a marked point. Combining tools from arithmetic geometry and holomorphic dynamics, they prove an 'unlikely intersection' statement for such pairs, thereby demonstrating strong rigidity features for them. They further describe one-dimensional families in the moduli space of polynomials containing infinitely many postcritically finite parameters, proving the dynamical André-Oort conjecture for curves in this context, originally stated by Baker and DeMarco.

MATHEMATICS / Geometry / Algebraic. --- Affine plane. --- Affine space. --- Affine transformation. --- Algebraic closure. --- Algebraic curve. --- Algebraic equation. --- Algebraic extension. --- Algebraic surface. --- Algebraic variety. --- Algebraically closed field. --- Analysis. --- Analytic function. --- Analytic geometry. --- Approximation. --- Arithmetic dynamics. --- Asymmetric graph. --- Ball (mathematics). --- Bifurcation theory. --- Boundary (topology). --- Cantor set. --- Characterization (mathematics). --- Chebyshev polynomials. --- Coefficient. --- Combinatorics. --- Complex manifold. --- Complex number. --- Computation. --- Computer programming. --- Conjugacy class. --- Connected component (graph theory). --- Continuous function (set theory). --- Coprime integers. --- Correspondence theorem (group theory). --- Counting. --- Critical graph. --- Cubic function. --- Datasheet. --- Disk (mathematics). --- Divisor (algebraic geometry). --- Elliptic curve. --- Equation. --- Equidistribution theorem. --- Equivalence relation. --- Euclidean topology. --- Existential quantification. --- Fixed point (mathematics). --- Function space. --- Geometric (company). --- Graph (discrete mathematics). --- Hamiltonian mechanics. --- Hausdorff dimension. --- Hausdorff measure. --- Holomorphic function. --- Inequality (mathematics). --- Instance (computer science). --- Integer. --- Intermediate value theorem. --- Intersection (set theory). --- Inverse-square law. --- Irreducible component. --- Iteration. --- Jordan curve theorem. --- Julia set. --- Limit of a sequence. --- Line (geometry). --- Metric space. --- Moduli space. --- Moment (mathematics). --- Montel's theorem. --- P-adic number. --- Parameter. --- Pascal's Wager. --- Periodic point. --- Polynomial. --- Power series. --- Primitive polynomial (field theory). --- Projective line. --- Quotient ring. --- Rational number. --- Realizability. --- Renormalization. --- Riemann surface. --- Ring of integers. --- Scientific notation. --- Set (mathematics). --- Sheaf (mathematics). --- Sign (mathematics). --- Stone–Weierstrass theorem. --- Subharmonic function. --- Support (mathematics). --- Surjective function. --- Theorem. --- Theory. --- Topology. --- Transfer principle. --- Union (set theory). --- Unit disk. --- Variable (computer science). --- Variable (mathematics). --- Zariski topology. --- Polynomials. --- Dynamics. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Algebra --- Algebraic geometry. --- Mathematics.