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Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

Algebraic geometry --- Topology --- Toposes --- Categories (Mathematics) --- Categories (Mathematics). --- Toposes. --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Category theory (Mathematics) --- Topoi (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Functor theory --- Adjoint functors. --- Associative property. --- Base change map. --- Base change. --- CW complex. --- Canonical map. --- Cartesian product. --- Category of sets. --- Category theory. --- Coequalizer. --- Cofinality. --- Coherence theorem. --- Cohomology. --- Cokernel. --- Commutative property. --- Continuous function (set theory). --- Contractible space. --- Coproduct. --- Corollary. --- Derived category. --- Diagonal functor. --- Diagram (category theory). --- Dimension theory (algebra). --- Dimension theory. --- Dimension. --- Enriched category. --- Epimorphism. --- Equivalence class. --- Equivalence relation. --- Existence theorem. --- Existential quantification. --- Factorization system. --- Functor category. --- Functor. --- Fundamental group. --- Grothendieck topology. --- Grothendieck universe. --- Group homomorphism. --- Groupoid. --- Heyting algebra. --- Higher Topos Theory. --- Higher category theory. --- Homotopy category. --- Homotopy colimit. --- Homotopy group. --- Homotopy. --- I0. --- Inclusion map. --- Inductive dimension. --- Initial and terminal objects. --- Inverse limit. --- Isomorphism class. --- Kan extension. --- Limit (category theory). --- Localization of a category. --- Maximal element. --- Metric space. --- Model category. --- Monoidal category. --- Monoidal functor. --- Monomorphism. --- Monotonic function. --- Morphism. --- Natural transformation. --- Nisnevich topology. --- Noetherian topological space. --- Noetherian. --- O-minimal theory. --- Open set. --- Power series. --- Presheaf (category theory). --- Prime number. --- Pullback (category theory). --- Pushout (category theory). --- Quillen adjunction. --- Quotient by an equivalence relation. --- Regular cardinal. --- Retract. --- Right inverse. --- Sheaf (mathematics). --- Sheaf cohomology. --- Simplicial category. --- Simplicial set. --- Special case. --- Subcategory. --- Subset. --- Surjective function. --- Tensor product. --- Theorem. --- Topological space. --- Topology. --- Topos. --- Total order. --- Transitive relation. --- Universal property. --- Upper and lower bounds. --- Weak equivalence (homotopy theory). --- Yoneda lemma. --- Zariski topology. --- Zorn's lemma.

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This computationally oriented work describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms.

Matrices. --- Numerical analysis. --- Mathematical analysis --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Matrices --- Numerical analysis --- Algorithm. --- Analysis of algorithms. --- Analytic function. --- Asymptotic analysis. --- Basis (linear algebra). --- Basis function. --- Biconjugate gradient method. --- Bidiagonal matrix. --- Bilinear form. --- Calculation. --- Characteristic polynomial. --- Chebyshev polynomials. --- Coefficient. --- Complex number. --- Computation. --- Condition number. --- Conjugate gradient method. --- Conjugate transpose. --- Cross-validation (statistics). --- Curve fitting. --- Degeneracy (mathematics). --- Determinant. --- Diagonal matrix. --- Dimension (vector space). --- Eigenvalues and eigenvectors. --- Equation. --- Estimation. --- Estimator. --- Exponential function. --- Factorization. --- Function (mathematics). --- Function of a real variable. --- Functional analysis. --- Gaussian quadrature. --- Hankel matrix. --- Hermite interpolation. --- Hessenberg matrix. --- Hilbert matrix. --- Holomorphic function. --- Identity matrix. --- Interlacing (bitmaps). --- Inverse iteration. --- Inverse problem. --- Invertible matrix. --- Iteration. --- Iterative method. --- Jacobi matrix. --- Krylov subspace. --- Laguerre polynomials. --- Lanczos algorithm. --- Linear differential equation. --- Linear regression. --- Linear subspace. --- Logarithm. --- Machine epsilon. --- Matrix function. --- Matrix polynomial. --- Maxima and minima. --- Mean value theorem. --- Meromorphic function. --- Moment (mathematics). --- Moment matrix. --- Moment problem. --- Monic polynomial. --- Monomial. --- Monotonic function. --- Newton's method. --- Numerical integration. --- Numerical linear algebra. --- Orthogonal basis. --- Orthogonal matrix. --- Orthogonal polynomials. --- Orthogonal transformation. --- Orthogonality. --- Orthogonalization. --- Orthonormal basis. --- Partial fraction decomposition. --- Polynomial. --- Preconditioner. --- QR algorithm. --- QR decomposition. --- Quadratic form. --- Rate of convergence. --- Recurrence relation. --- Regularization (mathematics). --- Rotation matrix. --- Singular value. --- Square (algebra). --- Summation. --- Symmetric matrix. --- Theorem. --- Tikhonov regularization. --- Trace (linear algebra). --- Triangular matrix. --- Tridiagonal matrix. --- Upper and lower bounds. --- Variable (mathematics). --- Vector space. --- Weight function.

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This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. This third edition contains a number of minor additions and improvements: A historical survey has been added, the definition of Lattés map has been made more inclusive, and the écalle-Voronin theory of parabolic points is described. The résidu itératif is studied, and the material on two complex variables has been expanded. Recent results on effective computability have been added, and the references have been expanded and updated. Written in his usual brilliant style, the author makes difficult mathematics look easy. This book is a very accessible source for much of what has been accomplished in the field.

Functions of complex variables --- Holomorphic mappings --- Riemann surfaces --- Fonctions d'une variable complexe --- Applications holomorphes --- Riemann, surfaces de --- Holomorphic mappings. --- Mappings, Holomorphic --- Functions of complex variables. --- Riemann surfaces. --- Surfaces, Riemann --- Functions --- Functions of several complex variables --- Mappings (Mathematics) --- Complex variables --- Elliptic functions --- Functions of real variables --- Absolute value. --- Addition. --- Algebraic equation. --- Attractor. --- Automorphism. --- Beltrami equation. --- Blaschke product. --- Boundary (topology). --- Branched covering. --- Coefficient. --- Compact Riemann surface. --- Compact space. --- Complex analysis. --- Complex number. --- Complex plane. --- Computation. --- Connected component (graph theory). --- Connected space. --- Constant function. --- Continued fraction. --- Continuous function. --- Coordinate system. --- Corollary. --- Covering space. --- Cross-ratio. --- Derivative. --- Diagram (category theory). --- Diameter. --- Diffeomorphism. --- Differentiable manifold. --- Disjoint sets. --- Disjoint union. --- Disk (mathematics). --- Division by zero. --- Equation. --- Euler characteristic. --- Existential quantification. --- Exponential map (Lie theory). --- Fundamental group. --- Harmonic function. --- Holomorphic function. --- Homeomorphism. --- Hyperbolic geometry. --- Inequality (mathematics). --- Integer. --- Inverse function. --- Irrational rotation. --- Iteration. --- Jordan curve theorem. --- Julia set. --- Lebesgue measure. --- Lecture. --- Limit point. --- Line segment. --- Linear map. --- Linearization. --- Mandelbrot set. --- Mathematical analysis. --- Maximum modulus principle. --- Metric space. --- Monotonic function. --- Montel's theorem. --- Normal family. --- Open set. --- Orbifold. --- Parameter space. --- Parameter. --- Periodic point. --- Point at infinity. --- Polynomial. --- Power series. --- Proper map. --- Quadratic function. --- Rational approximation. --- Rational function. --- Rational number. --- Real number. --- Riemann sphere. --- Riemann surface. --- Root of unity. --- Rotation number. --- Schwarz lemma. --- Scientific notation. --- Sequence. --- Simply connected space. --- Special case. --- Subgroup. --- Subsequence. --- Subset. --- Summation. --- Tangent space. --- Theorem. --- Topological space. --- Topology. --- Uniform convergence. --- Uniformization theorem. --- Unit circle. --- Unit disk. --- Upper half-plane. --- Winding number.

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Outer billiards is a basic dynamical system defined relative to a convex shape in the plane. B. H. Neumann introduced this system in the 1950's, and J. Moser popularized it as a toy model for celestial mechanics. All along, the so-called Moser-Neumann question has been one of the central problems in the field. This question asks whether or not one can have an outer billiards system with an unbounded orbit. The Moser-Neumann question is an idealized version of the question of whether, because of small disturbances in its orbit, the Earth can break out of its orbit and fly away from the Sun. In Outer Billiards on Kites, Richard Schwartz presents his affirmative solution to the Moser-Neumann problem. He shows that an outer billiards system can have an unbounded orbit when defined relative to any irrational kite. A kite is a quadrilateral having a diagonal that is a line of bilateral symmetry. The kite is irrational if the other diagonal divides the quadrilateral into two triangles whose areas are not rationally related. In addition to solving the basic problem, Schwartz relates outer billiards on kites to such topics as Diophantine approximation, the modular group, self-similar sets, polytope exchange maps, profinite completions of the integers, and solenoids--connections that together allow for a fairly complete analysis of the dynamical system.

Hyperbolic spaces. --- Singularities (Mathematics) --- Transformations (Mathematics) --- Geometry, Plane. --- Plane geometry --- Algorithms --- Differential invariants --- Geometry, Differential --- Geometry, Algebraic --- Hyperbolic complex manifolds --- Manifolds, Hyperbolic complex --- Spaces, Hyperbolic --- Geometry, Non-Euclidean --- Abelian group. --- Automorphism. --- Big O notation. --- Bijection. --- Binary number. --- Bisection. --- Borel set. --- C0. --- Calculation. --- Cantor set. --- Cartesian coordinate system. --- Combination. --- Compass-and-straightedge construction. --- Congruence subgroup. --- Conjecture. --- Conjugacy class. --- Continuity equation. --- Convex lattice polytope. --- Convex polytope. --- Coprime integers. --- Counterexample. --- Cyclic group. --- Diameter. --- Diophantine approximation. --- Diophantine equation. --- Disjoint sets. --- Disjoint union. --- Division by zero. --- Embedding. --- Equation. --- Equivalence class. --- Ergodic theory. --- Ergodicity. --- Factorial. --- Fiber bundle. --- Fibonacci number. --- Fundamental domain. --- Gauss map. --- Geometry. --- Half-integer. --- Homeomorphism. --- Hyperbolic geometry. --- Hyperplane. --- Ideal triangle. --- Intersection (set theory). --- Interval exchange transformation. --- Inverse function. --- Inverse limit. --- Isometry group. --- Lattice (group). --- Limit set. --- Line segment. --- Linear algebra. --- Linear function. --- Line–line intersection. --- Main diagonal. --- Modular group. --- Monotonic function. --- Multiple (mathematics). --- Orthant. --- Outer billiard. --- Parallelogram. --- Parameter. --- Partial derivative. --- Penrose tiling. --- Permutation. --- Piecewise. --- Polygon. --- Polyhedron. --- Polytope. --- Product topology. --- Projective geometry. --- Rectangle. --- Renormalization. --- Rhombus. --- Right angle. --- Rotational symmetry. --- Sanity check. --- Scientific notation. --- Semicircle. --- Sign (mathematics). --- Special case. --- Square root of 2. --- Subsequence. --- Summation. --- Symbolic dynamics. --- Symmetry group. --- Tangent. --- Tetrahedron. --- Theorem. --- Toy model. --- Translational symmetry. --- Trapezoid. --- Triangle group. --- Triangle inequality. --- Two-dimensional space. --- Upper and lower bounds. --- Upper half-plane. --- Without loss of generality. --- Yair Minsky.

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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.