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Hyperbolic Dynamics and Brownian Motion: An Introduction

Jacques Franchi and Yves Le Jan

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199654109
eISBN:
9780191745676
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199654109.001.0001
Subject:
Mathematics, Mathematical Physics

The idea of this book is to illustrate an interplay between distinct domains of mathematics. Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group PSO(1, d) ... More


A framework toward understanding the characterization of holomorphic dynamics

Yunping Jiang

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/9780691159294.003.0012
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter reviews the characterization of geometrically finite rational maps and then outlines a framework for characterizing holomorphic maps. Whereas Thurston's methods are based on estimates of ... More


Curved Space

Nicholas Manton and Nicholas Mee

in The Physical World: An Inspirational Tour of Fundamental Physics

Published in print:
2017
Published Online:
July 2017
ISBN:
9780198795933
eISBN:
9780191837111
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198795933.003.0006
Subject:
Physics, Condensed Matter Physics / Materials

This chapter develops the mathematical technology required to understand general relativity by taking the reader from the traditional flat space geometry of Euclid to the geometry of Riemann that ... More


Curves, Surfaces, and Hyperbolic Geometry

Benson Farb and Dan Margalit

in A Primer on Mapping Class Groups (PMS-49)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691147949
eISBN:
9781400839049
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691147949.003.0002
Subject:
Mathematics, Geometry / Topology

This chapter explains the basics of working with simple closed curves, focusing on the case of the closed surface Sɡ of genus g. When g is greater than or equal to 2, hyperbolic geometry enters as a ... More


Mappings and moduli

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/9780198526391.003.0014
Subject:
Mathematics, Geometry / Topology, Analysis

This chapter analyzes diffeomorphisms of the plane; braids, Dehn twists and quadratic singularities; hyperbolic geometry; and compactification of the moduli space.


Hyperbolic geometry

Jacques Franchi and Yves Le Jan

in Hyperbolic Dynamics and Brownian Motion: An Introduction

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199654109
eISBN:
9780191745676
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199654109.003.0002
Subject:
Mathematics, Mathematical Physics

Basic notions of hyperbolic geometry are presented: geodesics, light rays, tangent bundles etc. These are systematically derived from properties of Minkowski space: the boundary of hyperbolic space ... More


The Ping-Pong Lemma

Johanna Mangahas

in Office Hours with a Geometric Group Theorist

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691158662
eISBN:
9781400885398
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691158662.003.0005
Subject:
Mathematics, Geometry / Topology

This chapter considers an identifying feature of free groups: their ability to play ping-pong. In mathematics, you may encounter a group without immediately knowing which group it is. Fortunately, ... More


The Nielsen-Thurston Classification

Benson Farb and Dan Margalit

in A Primer on Mapping Class Groups (PMS-49)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691147949
eISBN:
9781400839049
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691147949.003.0014
Subject:
Mathematics, Geometry / Topology

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of ... More


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