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A Mathematical Gift, I: The interplay between topology, functions, geometry, and algebra
Kenji Ueno, Kyoto University, Japan, Koji Shiga, Yokohama, Japan, and Shigeyuki Morita, Tokyo Institute of Technology, Japan

Mathematical World
2003; 136 pp; softcover
Volume: 19
ISBN-10: 0-8218-3282-4
ISBN-13: 978-0-8218-3282-0
List Price: US$34
Member Price: US$27.20
Order Code: MAWRLD/19
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This item is also sold as part of the following set: MAWRLD-GSET

This book will bring the beauty and fun of mathematics to the classroom. It offers serious mathematics in a lively, reader-friendly style. Included are exercises and many figures illustrating the main concepts.

The first chapter presents the geometry and topology of surfaces. Among other topics, the authors discuss the Poincaré-Hopf theorem on critical points of vector fields on surfaces and the Gauss-Bonnet theorem on the relation between curvature and topology (the Euler characteristic). The second chapter addresses various aspects of the concept of dimension, including the Peano curve and the Poincaré approach. Also addressed is the structure of three-dimensional manifolds. In particular, it is proved that the three-dimensional sphere is the union of two doughnuts.

This is the first of three volumes originating from a series of lectures given by the authors at Kyoto University (Japan). It is suitable for classroom use for high school mathematics teachers and undergraduate mathematics courses in the sciences and liberal arts. The second volume is available as Volume 20 in the AMS series, Mathematical World. A third volume is forthcoming.


Advanced high-school students and undergraduates in mathematics.


"With A Mathematical Gift, I, there is no reason why every undergraduate student should not be exposed to some topology ... accessible to even high school students ... beautiful illustrations and straightforward explanations of sophisticated ideas. Real world and concrete scenarios are used ... elegant explanations ... exercises ... are friendly and non-threatening ... the perfect choice for anyone who conducts summer workshops for high school students ... an ideal supplement for graduate students studying topology for the first time ... also excellent for undergraduate independent studies ... I enjoyed reading this book ... fun to look at ... instructive and motivating ... impressed ... wowed by the detail and clarity presented by the authors. Readers of A Mathematical Gift, I will want to read A Mathematical Gift, II."

-- MAA Online

Table of Contents

Invitation to topology (Viewing figures globally)
  • Introduction
  • The Euler characteristic
  • Vortices created by winds and the Euler characteristic
  • Curvature of a surface and the Euler characteristic
The story of dimension
  • Introduction
  • Learning to appreciate dimension
  • What is dimension?
  • Three-dimensional figures
  • Physics and dimension
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