Softcover ISBN:  9780821848166 
Product Code:  STML/49 
List Price:  $55.00 
Individual Price:  $44.00 
Electronic ISBN:  9781470416348 
Product Code:  STML/49.E 
List Price:  $54.00 
Individual Price:  $43.20 

Book DetailsStudent Mathematical LibraryIAS/Park City Mathematics SubseriesVolume: 49; 2009; 384 ppMSC: Primary 51; 30; 57;
The study of 3dimensional spaces brings together elements from several areas of mathematics. The most notable are topology and geometry, but elements of number theory and analysis also make appearances. In the past 30 years, there have been striking developments in the mathematics of 3dimensional manifolds. This book aims to introduce undergraduate students to some of these important developments.
LowDimensional Geometry starts at a relatively elementary level, and its early chapters can be used as a brief introduction to hyperbolic geometry. However, the ultimate goal is to describe the very recently completed geometrization program for 3dimensional manifolds. The journey to reach this goal emphasizes examples and concrete constructions as an introduction to more general statements. This includes the tessellations associated to the process of gluing together the sides of a polygon. Bending some of these tessellations provides a natural introduction to 3dimensional hyperbolic geometry and to the theory of kleinian groups, and it eventually leads to a discussion of the geometrization theorems for knot complements and 3dimensional manifolds.
This book is illustrated with many pictures, as the author intended to share his own enthusiasm for the beauty of some of the mathematical objects involved. However, it also emphasizes mathematical rigor and, with the exception of the most recent research breakthroughs, its constructions and statements are carefully justified.This book is published in cooperation with IAS/Park City Mathematics Institute.ReadershipUndergraduate students interested in topology and/or geometry of lowdimensional manifolds, particularly 3manifolds.

Table of Contents

Chapters

Chapter 1. The euclidean plane

Chapter 2. The hyperbolic plane

Chapter 3. The 2dimensional sphere

Chapter 4. Gluing constructions

Chapter 5. Gluing examples

Chapter 6. Tessellations

Chapter 7. Group actions and fundamental domains

Chapter 8. The Farey tessellation and circle packing

Chapter 9. The 3dimensional hyperbolic space

Chapter 10. Kleinian groups

Chapter 11. The figureeight knot complement

Chapter 12. Geometrization theorems in dimension 3

Appendix. Tool kit


Additional Material

Reviews

...an essential book, if just because no other yet competes for this crucial niche, but also happily excellent in every respect—passionately told, expertly rendered, exquisitely organized, and sumptuously illustrated. ... Essential.
CHOICE Magazine


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The study of 3dimensional spaces brings together elements from several areas of mathematics. The most notable are topology and geometry, but elements of number theory and analysis also make appearances. In the past 30 years, there have been striking developments in the mathematics of 3dimensional manifolds. This book aims to introduce undergraduate students to some of these important developments.
LowDimensional Geometry starts at a relatively elementary level, and its early chapters can be used as a brief introduction to hyperbolic geometry. However, the ultimate goal is to describe the very recently completed geometrization program for 3dimensional manifolds. The journey to reach this goal emphasizes examples and concrete constructions as an introduction to more general statements. This includes the tessellations associated to the process of gluing together the sides of a polygon. Bending some of these tessellations provides a natural introduction to 3dimensional hyperbolic geometry and to the theory of kleinian groups, and it eventually leads to a discussion of the geometrization theorems for knot complements and 3dimensional manifolds.
This book is illustrated with many pictures, as the author intended to share his own enthusiasm for the beauty of some of the mathematical objects involved. However, it also emphasizes mathematical rigor and, with the exception of the most recent research breakthroughs, its constructions and statements are carefully justified.
Undergraduate students interested in topology and/or geometry of lowdimensional manifolds, particularly 3manifolds.

Chapters

Chapter 1. The euclidean plane

Chapter 2. The hyperbolic plane

Chapter 3. The 2dimensional sphere

Chapter 4. Gluing constructions

Chapter 5. Gluing examples

Chapter 6. Tessellations

Chapter 7. Group actions and fundamental domains

Chapter 8. The Farey tessellation and circle packing

Chapter 9. The 3dimensional hyperbolic space

Chapter 10. Kleinian groups

Chapter 11. The figureeight knot complement

Chapter 12. Geometrization theorems in dimension 3

Appendix. Tool kit

...an essential book, if just because no other yet competes for this crucial niche, but also happily excellent in every respect—passionately told, expertly rendered, exquisitely organized, and sumptuously illustrated. ... Essential.
CHOICE Magazine