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This book introduces the theory of complex surfaces through a comprehensive look at finite covers of the projective plane branched along line arrangements. Paula Tretkoff emphasizes those finite covers that are free "ients of the complex two-dimensional ball. Tretkoff also includes background on the classical Gauss hypergeometric function of one variable, and a chapter on the Appell two-variable F1 hypergeometric function.The material in this book began as a set of lecture notes, taken by Tretkoff, of a course given by Friedrich Hirzebruch at ETH Zürich in 1996. The lecture notes were then considerably expanded by Hirzebruch and Tretkoff over a number of years. In this book, Tretkoff has expanded those notes even further, still stressing examples offered by finite covers of line arrangements. The book is largely self-contained and foundational material is introduced and explained as needed, but not treated in full detail. References to omitted material are provided for interested readers.Aimed at graduate students and researchers, this is an accessible account of a highly informative area of complex geometry.

Curves, Elliptic. --- Geometry, Algebraic. --- Projective planes. --- Unit ball. --- Riemann surfaces. --- Surfaces, Riemann --- Functions --- Ball, Unit --- Holomorphic functions --- Planes, Projective --- Geometry, Projective --- Algebraic geometry --- Geometry --- Elliptic curves --- Curves, Algebraic --- Appell hypergeometric function. --- Chern numbers. --- Euler number. --- Friedrich Hirzebruch. --- Gauss hypergeometric function. --- Gaussian curvature. --- Hermitian metric. --- Kodaira dimension. --- Mbius transformation. --- Miyaoka-Yau inequality. --- Riemann surface. --- Riemannian metric. --- algebraic geometry. --- algebraic surface. --- arithmetic monodromy group. --- b-space. --- ball "ient. --- canonical divisor class. --- complete quadrilateral. --- complex 2-ball. --- complex manifold. --- complex surface. --- covering group. --- covering space. --- differential geometry. --- divisor class group. --- divisor. --- elliptic curve. --- finite covering. --- first Chern class. --- fractional linear transformation. --- free 2-ball "ient. --- fundamental group. --- geometry. --- intersection point. --- line arrangement. --- line bundle. --- linear arrangement. --- log-canonical divisor. --- minimal surface. --- monodromy group. --- orbifold structure. --- orbifold. --- partial differential equation. --- plurigenus. --- projective plane. --- proportionality deviation. --- ramification indices. --- rational curve. --- regular point. --- signature. --- solution space. --- topological invariant. --- transversely intersecting divisor. --- triangle groups. --- weight.

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The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is, twists of hypergeometric functions in n-variables. These are treated as an (n+1) dimensional vector space of multivalued locally holomorphic functions defined on the space of n+3 tuples of distinct points on the projective line P modulo, the diagonal section of Auto P=m. For n=1, the characterization may be regarded as a generalization of Riemann's classical theorem characterizing hypergeometric functions by their exponents at three singular points. This characterization permits the authors to compare monodromy groups corresponding to different parameters and to prove commensurability modulo inner automorphisms of PU(1,n). The book includes an investigation of elliptic and parabolic monodromy groups, as well as hyperbolic monodromy groups. The former play a role in the proof that a surprising number of lattices in PU(1,2) constructed as the fundamental groups of compact complex surfaces with constant holomorphic curvature are in fact conjugate to projective monodromy groups of hypergeometric functions. The characterization of hypergeometric-like functions by their exponents at the divisors "at infinity" permits one to prove generalizations in n-variables of the Kummer identities for n-1 involving quadratic and cubic changes of the variable.

Hypergeometric functions. --- Monodromy groups. --- Lattice theory. --- Abuse of notation. --- Algebraic variety. --- Analytic continuation. --- Arithmetic group. --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Codimension. --- Coefficient. --- Cohomology. --- Commensurability (mathematics). --- Compactification (mathematics). --- Complete quadrangle. --- Complex number. --- Complex space. --- Conjugacy class. --- Connected component (graph theory). --- Coprime integers. --- Cube root. --- Derivative. --- Diagonal matrix. --- Differential equation. --- Dimension (vector space). --- Discrete group. --- Divisor (algebraic geometry). --- Divisor. --- Eigenvalues and eigenvectors. --- Ellipse. --- Elliptic curve. --- Equation. --- Existential quantification. --- Fiber bundle. --- Finite group. --- First principle. --- Fundamental group. --- Gelfand. --- Holomorphic function. --- Hypergeometric function. --- Hyperplane. --- Hypersurface. --- Integer. --- Inverse function. --- Irreducible component. --- Irreducible representation. --- Isolated point. --- Isomorphism class. --- Line bundle. --- Linear combination. --- Linear differential equation. --- Local coordinates. --- Local system. --- Locally finite collection. --- Mathematical proof. --- Minkowski space. --- Moduli space. --- Monodromy. --- Morphism. --- Multiplicative group. --- Neighbourhood (mathematics). --- Open set. --- Orbifold. --- Permutation. --- Picard group. --- Point at infinity. --- Polynomial ring. --- Projective line. --- Projective plane. --- Projective space. --- Root of unity. --- Second derivative. --- Simple group. --- Smoothness. --- Subgroup. --- Subset. --- Symmetry group. --- Tangent space. --- Tangent. --- Theorem. --- Transversal (geometry). --- Uniqueness theorem. --- Variable (mathematics). --- Vector space.

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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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Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics. Its central concern is the structure of an infinitely renormalizable quadratic polynomial f(z) = z2 + c. As discovered by Feigenbaum, such a mapping exhibits a repetition of form at infinitely many scales. Drawing on universal estimates in hyperbolic geometry, this work gives an analysis of the limiting forms that can occur and develops a rigidity criterion for the polynomial f. This criterion supports general conjectures about the behavior of rational maps and the structure of the Mandelbrot set. The course of the main argument entails many facets of modern complex dynamics. Included are foundational results in geometric function theory, quasiconformal mappings, and hyperbolic geometry. Most of the tools are discussed in the setting of general polynomials and rational maps.

Renormalization (Physics) --- Polynomials. --- Dynamics. --- Mathematical physics. --- Analytic function. --- Attractor. --- Automorphism. --- Bernhard Riemann. --- Bounded set. --- Branched covering. --- Cantor set. --- Cardioid. --- Chain rule. --- Coefficient. --- Combinatorics. --- Complex manifold. --- Complex plane. --- Complex torus. --- Conformal geometry. --- Conformal map. --- Conjecture. --- Connected space. --- Covering space. --- Cyclic group. --- Degeneracy (mathematics). --- Dense set. --- Diagram (category theory). --- Diameter. --- Differential geometry of surfaces. --- Dihedral group. --- Dimension (vector space). --- Dimension. --- Disjoint sets. --- Disk (mathematics). --- Dynamical system. --- Endomorphism. --- Equivalence class. --- Equivalence relation. --- Ergodic theory. --- Euler characteristic. --- Filled Julia set. --- Geometric function theory. --- Geometry. --- Hausdorff dimension. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Hyperbolic geometry. --- Implicit function theorem. --- Injective function. --- Integer matrix. --- Interval (mathematics). --- Inverse limit. --- Julia set. --- Kleinian group. --- Limit point. --- Limit set. --- Linear map. --- Mandelbrot set. --- Manifold. --- Markov partition. --- Mathematical induction. --- Maxima and minima. --- Measure (mathematics). --- Moduli (physics). --- Monic polynomial. --- Montel's theorem. --- Möbius transformation. --- Natural number. --- Open set. --- Orbifold. --- Periodic point. --- Permutation. --- Point at infinity. --- Pole (complex analysis). --- Polynomial. --- Proper map. --- Quadratic differential. --- Quadratic function. --- Quadratic. --- Quasi-isometry. --- Quasiconformal mapping. --- Quotient space (topology). --- Removable singularity. --- Renormalization. --- Riemann mapping theorem. --- Riemann sphere. --- Riemann surface. --- Rigidity theory (physics). --- Scalar (physics). --- Schwarz lemma. --- Scientific notation. --- Special case. --- Structural stability. --- Subgroup. --- Subsequence. --- Symbolic dynamics. --- Tangent space. --- Theorem. --- Uniformization theorem. --- Uniformization. --- Union (set theory). --- Unit disk. --- Upper and lower bounds.