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Geometry with an Introduction to Cosmic Topology

Author(s): Michael P. Hitchman, Linfield College
Details:
  • ISBN-13: 9780763754570
  • Hardcover    238 pages      © 2009
Price: International Sales $118.95 US List
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Overview

Geometry with an Introduction to Cosmic Topology is motivated by questions that have ignited the imagination of stargazers since antiquity.  What is the shape of the universe? Does the universe have an edge?  Is it infinitely big? Dr. Hitchman aims to clarify this fascinating area of mathematics and focuses on the mathematical tools used to investigate the shape of the universe. The text follows the Erlangen Program, which develops geometry in terms of a space and a group of transformations of that space. This approach to non-Euclidean geometry provides excellent material by which students can learn the more sophisticated modes of thinking necessary in upper-division mathematics courses.

This unique text is organized into three natural parts:

Chapter 1 introduces the geometric perspective taken in the text and the motivation for the material that comes from cosmology.

Chapters 2-7 contain the core mathematical content of the text, developing hyperbolic elliptic, and Euclidean geometry from the complex plane and subgroups of Mobius transformations. Other topics include the topology and geometry of surfaces and Dirichlet domains.

Finally, Chapter 8 explores the topic of cosmic topology through the geometry learned in the preceding chapters.

ShowKey Features

The approach to geometry is that of the Erlangen Program. This approach draws on content and uses modes of thought that appear in advanced courses in the undergraduate mathematics curriculum. 

The use of constructions to prove theorems and motivate definitions gives a tangible and intuitive approach to the subject.  The constructive approach adopted in the text enables the students to connect more deeply with the material. 

Includes 164 illustrations to aid visualization, and several applets for manipulating figures in non-Euclidean geometry.

The exercises come in many varieties to help students practice different skills: intense computations (e.g., computing hyperbolic distances, the hyperbolic law of cosines), constructions (e.g., symmetry points, hyperbolic circles), proofs both big (e.g., proof of arc length invariance) and small (that a collection of transformations forms a group), open-ended exercises (do you think the pair separation histogram came from a toroidal universe or a Klein bottle universe?), applications of geometry and topology to cosmology (Sections 7.4, 7.6, 8.2), and investigative.

Chapter 8 includes a general discussion of two current research programs in cosmic topology: cosmic crystallography and circles-in-the-sky. Chapter 8 also includes a review of our understanding of the current state of the universe, a view that is based on significant observational discoveries over the last 10 years. 

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ShowTable of Contents

Preface  
Chapter 1  An Invitation to Geometry
Chapter 2  The Plane and Complex Numbers
Chapter 3  Transformations
Chapter 4  Geometry
Chapter 5  Hyperbolic Geometry
Chapter 6  Elliptic Geometry
Chapter 7  Geometry on Surfaces
Chapter 8  Cosmic Topology
List of Notation  
Bibliography  
Index  
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ShowAbout the Author(s)

Michael P. Hitchman-Linfield College

Dr. Hitchman earned a B.A. in mathematics from Swarthmore College and a PhD at the University of Oregon.  He has taught mathematics at Lewis & Clark College, The College of Idaho, and most recently Linfield College.  His mathematical interests include low dimensional topology and cosmic topology.

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ShowAppropriate Courses

Appropriate for 300 and 400 level geometry courses for mathematics majors.

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ShowResources

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