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Riemann Surfaces, Coverings, and Hypergeometric Functions

Paula Tretkoff

in Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)

This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and ... More

This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and notes that an automorphism of a Riemann surface is a biholomorphic map of the Riemann surface onto itself. It then describes a Riemannian metric and the Gauss-Bonnet theorem, which can be interpreted as a relation between the Gaussian curvature of a compact Riemann surface X and its Euler characteristic. It also examines the behavior of the Euler number under finite covering, along with finite subgroups of the group of fractional linear transformations PSL(2, C). Finally, it presents some basic facts about the classical Gauss hypergeometric functions of one complex variable, triangle groups acting discontinuously on one of the simply connected Riemann surfaces, and the hypergeometric monodromy group.Less

THE VIRTUES OF CRACKED REASONING

Mark Wilson

in Wandering Significance: An Essay on Conceptual Behaviour

Drawing upon the rich experience gathered within applied mathematics, various ‘facade patterns’ are examined that frequently develop when an originating usage enlarges its descriptive scope through ... More

Drawing upon the rich experience gathered within applied mathematics, various ‘facade patterns’ are examined that frequently develop when an originating usage enlarges its descriptive scope through patch-to-patch prolongation. The completed results can generate a global structure that is syntactically inconsistent as a whole, yet avoids logical ruination through simple restrictions upon data exportation from one patch to another (a ‘Riemann surface’ represents a standard mathematical prototype of the phenomenon). It is argued that not only do such facades often represent the natural end products of ordinary linguistic development, they often provide particularly effective forms of linguistic engineering. Philosophical puzzles sometimes arise when these alternative patterns of semantic design get mistaken for classical models, as the troubled history of ‘force’ effectively illustrates. The fact that we can rarely determine whether an initial collection of descriptive vocabulary is destined to develop into a facade rather than implementing a simpler pattern of word/world alignment provides a convenient indication of the degree to which a classical picture of conceptual grasp exaggerates our capacity to augur the fate of our descriptive words over time.Less

Multivalued harmonic morphisms

Paul Baird and John C. Wood

in Harmonic Morphisms Between Riemannian Manifolds

Multivalued analytic functions are considered to be functions on a Riemann surface, which can be defined either by cutting and pasting or by a graph construction. Multivalued maps in general are ... More

Multivalued analytic functions are considered to be functions on a Riemann surface, which can be defined either by cutting and pasting or by a graph construction. Multivalued maps in general are considered, and it is shown how the concept of multivalued analytic functions generalizes to harmonic morphisms using the graph construction. An alternative treatment for space forms is presented, and some specific examples are given. The chapter concludes with a discussion of the behaviour on the branching set of the projection map for a multivalued harmonic morphism on a three-dimensional space form.Less

Holomorphic harmonic morphisms

Paul Baird and John C. Wood

in Harmonic Morphisms Between Riemannian Manifolds

The first section of this chapter investigates the general question of when holomorphic maps between almost Hermitian manifolds are harmonic maps or harmonic morphisms. In particular, a holomorphic ... More

The first section of this chapter investigates the general question of when holomorphic maps between almost Hermitian manifolds are harmonic maps or harmonic morphisms. In particular, a holomorphic map from a Kaehler manifold to a Riemann surface is always a harmonic morphism. It is shown how harmonic morphisms into a Riemann surface can sometimes be combined to give new ones. The construction of harmonic morphisms from domains of Euclidean and related spaces is discussed, which are holomorphic with respect to a Hermitian structure, thus finding interesting globally defined complex-valued harmonic morphisms on a Euclidean space of arbitrary dimension.Less

Meromorphic functions and the Main Theorem for compact Riemann surfaces

Simon Donaldson

in Riemann Surfaces

This chapter explains a strategy for proving the fundamental structural results about Riemann surfaces. It first considers the consequence of the Main Theorem for compact Riemann surfaces. It then ... More

This chapter explains a strategy for proving the fundamental structural results about Riemann surfaces. It first considers the consequence of the Main Theorem for compact Riemann surfaces. It then analyzes the Riemann-Roch formula, the fundamental tool in the theory of compact Riemann surfaces.Less

TOPOLOGICAL STRINGS

Marcos Mariño

in Chern-Simons Theory, Matrix Models, and Topological Strings

Type-A and type-B topological sigma models are two topological field theories in two dimensions. Although they contain a lot of information in genus 0, they turn out to be trivial for g > 1. This is ... More

Type-A and type-B topological sigma models are two topological field theories in two dimensions. Although they contain a lot of information in genus 0, they turn out to be trivial for g > 1. This is essentially due to the fact that, in order to define these theories, it is necessary to consider a fixed metric in the Riemann surface. In order to obtain a non-trivial theory in higher genus the degrees of freedom of the two-dimensional metric must be introduced. This means that the topological field theories must be coupled to two-dimensional gravity. The coupling to gravity is done by using the fact that the structure of the twisted theory is tantalizingly close to that of the bosonic string. Topological sigma models may be defined not only on closed Riemann surfaces and closed topological strings, but also on the open case.Less

Contrasts in Riemann surface theory

Simon Donaldson

in Riemann Surfaces

Section 4.2.3 showed how to associate a compact connected Riemann surface to any irreducible polynomial P(z,w) in two variables. This chapter first establishes the converse — any compact connected ... More

Section 4.2.3 showed how to associate a compact connected Riemann surface to any irreducible polynomial P(z,w) in two variables. This chapter first establishes the converse — any compact connected Riemann surface arises from a polynomial P. It also explains how the whole theory can be cast in an algebraic form, involving fields of transcendence degree 1. The chapter then analyzes hyperbolic surfaces, covering models of the hyperbolic plane; self-isometries; geodesics; the Gauss-Bonnet Theorem; right-angled hexagons; and closed geodesics.Less

Facades

Christopher Pincock

in Mathematics and Scientific Representation

Mark Wilson’s book Wandering Significance: An Essay on Conceptual Behavior is considered in this chapter. Wilson argues that common views on mathematical and scientific concepts distort our ... More

Mark Wilson’s book Wandering Significance: An Essay on Conceptual Behavior is considered in this chapter. Wilson argues that common views on mathematical and scientific concepts distort our understanding of scientific knowledge and representation. Pincock substantially agrees with Wilson and describes how the positions from earlier chapters can be adapted to fit with Wilson’s worries. At the heart of Wilson’s worries is a view of representations as made up of a collection of locally effective patches. Pincock explains how mathematics generates these patches and helps to link them together. The example of the development of the concept of Riemann surfaces is used to further clarify this picture of scientific representation and to vindicate a form of patient scientific realism.Less

Basic definitions

Simon Donaldson

in Riemann Surfaces

This chapter presents definitions of Riemann surfaces and holomorphic maps, and provides examples, including algebraic curves and quotients.

Functions of a complex variable

Chun Wa Wong

in Introduction to Mathematical Physics: Methods & Concepts

Functions of a complex variable are shown to be more complete and rigid than functions of a real variable. Analytic functions with well defined derivatives satisfy two Cauch–Riemann conditions. ... More

Functions of a complex variable are shown to be more complete and rigid than functions of a real variable. Analytic functions with well defined derivatives satisfy two Cauch–Riemann conditions. Multivalued functions can be made single-valued on a multi-sheet Riemann surface. The values of an analytic function in a region of the complex plane are completely defined by the knowledge of their values on a closed boundary of the region. Important properties and techniques of complex analysis are described. These include Taylor and Laurent expansions, contour integration and residue calculus, Green's functions, Laplace transforms and Bromwich integrals, dispersion relations and asymptotic expansions. Analytic functions are defined by their properties at the locations called singularities (poles and branch cuts) where they cease to be analytic. This feature makes analytic functions of particular interest in the construction of physical theories.Less

Elliptic functions and integrals

Simon Donaldson

in Riemann Surfaces

This chapter examines Riemann surfaces of genus 1. The constructions give an important model for the more general theory to be developed in Part III. The constructions also involve classical topics ... More

This chapter examines Riemann surfaces of genus 1. The constructions give an important model for the more general theory to be developed in Part III. The constructions also involve classical topics in mathematics, which relate the abstractions of Riemann surface theory to their origin in concrete calculus problems.Less

Calculus on surfaces

Simon Donaldson

in Riemann Surfaces

This chapter develops the theory of differential forms on smooth surfaces and Riemann surfaces. The analysis begins with smooth surfaces, covering cotangent spaces and 1-forms; 2-forms and ... More

This chapter develops the theory of differential forms on smooth surfaces and Riemann surfaces. The analysis begins with smooth surfaces, covering cotangent spaces and 1-forms; 2-forms and integration. It then turns to de Rham cohomology, which includes cohomology with compact support, and Poincaré duality. The final section discusses calculus on Riemann surfaces, covering decomposition of the 1-forms; the Laplace operator and harmonic functions; and the Dirichlet norm.Less

Entropy for hyperbolic Riemann surface laminations I

Tien-Cuong Dinh, Viet-Anh Nguyen, and Nessim Sibony

Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)

in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

This chapter introduces a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. It also studies the transverse regularity of the Poincaré metric and the finiteness of ... More

This chapter introduces a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. It also studies the transverse regularity of the Poincaré metric and the finiteness of the entropy. The chapter first focuses on compact laminations, which are transversally smooth, before turning to the case of singular foliations, showing how the Poincaré metric on leaves is transversally Hölder continuous. In addition, the chapter considers the problem in the proof that the entropy is finite for singular foliations is quite delicate and requires a careful analysis of the dynamics around the singularities. Finally, the chapter discusses a notion of metric entropy for harmonic probability measures and gives some open questions.Less

Dedekind on Continuity

Emmylou Haffner and Dirk Schlimm

in The History of Continua: Philosophical and Mathematical Perspectives

In this chapter we present Richard Dedekind’s conception of continuity and his various approaches to continuous domains in a historical context. In addition to his seminal work on foundations of ... More

In this chapter we present Richard Dedekind’s conception of continuity and his various approaches to continuous domains in a historical context. In addition to his seminal work on foundations of irrational numbers (Stetigkeit und irrationale Zahlen, 1872), we also include a discussion of more mathematical texts (both published and unpublished) in which Dedekind also treats other continuous domains, such as Riemann surfaces, spaces, and multiply extended continuous domains. Dedekind’s reflections on these matters illustrate the wide range and general coherence of his thoughts. In particular, while Dedekind’s approach to mathematics can be characterized as being axiomatic, mapping-based, structuralist, and increasingly abstract, we argue that there is also a more general outlook underlying his methodology, which can be described as being, broadly understood, arithmetical.Less

The Uniformisation Theorem

Simon Donaldson

in Riemann Surfaces

This chapter proves the following theorem. Theorem 12: Let X be a connected, simply connected, non-compact Riemann surface. Then X is equivalent to either C or the upper half-plane H. The proof ... More

This chapter proves the following theorem. Theorem 12: Let X be a connected, simply connected, non-compact Riemann surface. Then X is equivalent to either C or the upper half-plane H. The proof presented here follows the same general pattern as one already given to classify compact simply connected Riemann surfaces, but the non-compactness will require some extra steps.Less

Holomorphic functions

Simon Donaldson

in Riemann Surfaces

This chapter reviews some examples of holomorphic functions in complex analysis. It emphasizes the idea of ‘analytic continuation’, which is a fundamental motivation for Riemann surface theory.

Entropy for hyperbolic Riemann surface laminations II

Tien-Cuong Dinh, Viet-Anh Nguyen, and Nessim Sibony

Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)

in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

This chapter studies Riemann surface foliations with tame singular points. It shows that the hyperbolic entropy of a Brody hyperbolic foliation by Riemann surfaces with linearizable isolated ... More

This chapter studies Riemann surface foliations with tame singular points. It shows that the hyperbolic entropy of a Brody hyperbolic foliation by Riemann surfaces with linearizable isolated singularities on a compact complex surface is finite. The chapter then proves the finiteness of the entropy in the local setting near a singular point in any dimension, using a division of a neighborhood of a singular point into adapted cells. Next, the chapter estimates the modulus of continuity for the Poincaré metric along the leaves of the foliation, using notion of conformally (R,δ‎)-close maps. The estimate holds for foliations on manifolds of higher dimension.Less

Proof of the Main Theorem

Simon Donaldson

in Riemann Surfaces

This chapter provides proof of the main analytical result, Theorem 5, for compact Riemann surfaces.

Parabolic Higgs Bundles for Real Reductive Lie Groups: A Very Basic Introduction

Ignasi Mundet i Riera

in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

This chapter explains the correspondence between local systems on a punctured Riemann surface with the structure group being a real reductive Lie group G, and parabolic G-Higgs bundles. The chapter ... More

This chapter explains the correspondence between local systems on a punctured Riemann surface with the structure group being a real reductive Lie group G, and parabolic G-Higgs bundles. The chapter describes the objects involved in this correspondence, taking some time to motivate them by recalling the definitions of G-Higgs bundles without parabolic structure and of parabolic vector bundles. Finally, it explains the relevant polystability condition and the correspondence between local systems and Higgs bundles.Less