## Riemann Surfaces, Coverings, and Hypergeometric Functions

*Paula Tretkoff*

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

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0003
- Subject:
- Mathematics, Geometry / Topology

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

## THE VIRTUES OF CRACKED REASONING

*Mark Wilson*

### in Wandering Significance: An Essay on Conceptual Behaviour

- Published in print:
- 2006
- Published Online:
- January 2007
- ISBN:
- 9780199269259
- eISBN:
- 9780191710155
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199269259.003.0006
- Subject:
- Philosophy, Philosophy of Language

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

## Multivalued harmonic morphisms

*Paul Baird and John C. Wood*

### in Harmonic Morphisms Between Riemannian Manifolds

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0009
- Subject:
- Mathematics, Pure Mathematics

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

## Holomorphic harmonic morphisms

*Paul Baird and John C. Wood*

### in Harmonic Morphisms Between Riemannian Manifolds

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0008
- Subject:
- Mathematics, Pure Mathematics

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

## Meromorphic functions and the Main Theorem for compact Riemann surfaces

*Simon Donaldson*

### in Riemann Surfaces

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0008
- Subject:
- Mathematics, Geometry / Topology, Analysis

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

## TOPOLOGICAL STRINGS

*Marcos Mariño*

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

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568490
- eISBN:
- 9780191717604
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568490.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

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

## Riemann Surfaces

*Simon Donaldson*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.001.0001
- Subject:
- Mathematics, Geometry / Topology, Analysis

The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an ... More

## Contrasts in Riemann surface theory

*Simon Donaldson*

### in Riemann Surfaces

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0011
- Subject:
- Mathematics, Geometry / Topology, Analysis

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

## Facades

*Christopher Pincock*

### in Mathematics and Scientific Representation

- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780199757107
- eISBN:
- 9780199932313
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199757107.003.0013
- Subject:
- Philosophy, Philosophy of Science

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

## Basic definitions

*Simon Donaldson*

### in Riemann Surfaces

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0003
- Subject:
- Mathematics, Geometry / Topology, Analysis

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

- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199641390
- eISBN:
- 9780191747786
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199641390.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

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

## Elliptic functions and integrals

*Simon Donaldson*

### in Riemann Surfaces

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0006
- Subject:
- Mathematics, Geometry / Topology, Analysis

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

## Calculus on surfaces

*Simon Donaldson*

### in Riemann Surfaces

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0005
- Subject:
- Mathematics, Geometry / Topology, Analysis

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

## 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

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0020
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

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

## Dedekind on Continuity

*Emmylou Haffner and Dirk Schlimm*

### in The History of Continua: Philosophical and Mathematical Perspectives

- Published in print:
- 2020
- Published Online:
- December 2020
- ISBN:
- 9780198809647
- eISBN:
- 9780191846915
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198809647.003.0011
- Subject:
- Philosophy, Metaphysics/Epistemology

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

## The Uniformisation Theorem

*Simon Donaldson*

### in Riemann Surfaces

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0010
- Subject:
- Mathematics, Geometry / Topology, Analysis

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

## Holomorphic functions

*Simon Donaldson*

### in Riemann Surfaces

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0001
- Subject:
- Mathematics, Geometry / Topology, Analysis

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

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0021
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

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

## Proof of the Main Theorem

*Simon Donaldson*

### in Riemann Surfaces

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0009
- Subject:
- Mathematics, Geometry / Topology, Analysis

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

- Published in print:
- 2018
- Published Online:
- December 2018
- ISBN:
- 9780198802020
- eISBN:
- 9780191869068
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198802020.003.0027
- Subject:
- Mathematics, Geometry / Topology

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