A Mathematical Gift, I: The interplay between topology, functions, geometry, and algebra
About this Title
Kenji Ueno, Kyoto University, Kyoto, Japan, Koji Shiga, , Yokohama, Japan and Shigeyuki Morita, Tokyo Institute of Technology, Tokyo, Japan. Translated by Eiko Tyler
Publication: Mathematical World
Publication Year 2003: Volume 19
ISBN: 978-0-8218-3282-0 (print)
MathSciNet review: MR2013039
MSC: Primary 57-01; Secondary 53-01
This volume is not part of this online collection, but can be purchased through our online bookstore.
This is the first of three volumes originated from a series of lectures in mathematics given by professors of Kyoto University in Japan for high school students. The main purpose of the lectures was to show the listeners the beauty and liveliness of mathematics using the material that is accessible to people with little preliminary knowledge.
The first chapter of the book talks about the geometry and topology of surfaces. Among other topics the authors discuss the Poincaré–Hopf theorem about critical points of vector fields on surfaces and the Gauss–Bonnet theorem about the relation between the curvature and topology (Euler characteristics). The second chapter addresses various aspects of the concept of dimension, including the Peano curve and the Poincaré approach to dimension. It also discusses the structure of three-dimensional manifolds, proving, in particular, that the three-dimensional sphere is the union of two doughnuts.
Advanced high-school students and undergraduates in mathematics.
Table of Contents
Invitation to topology (Viewing figures globally)
- 1. Introduction
- 2. The Euler characteristic
- 3. Vortices created by winds and the Euler characteristic
- 4. Curvature of a surface and the Euler characteristic
The story of dimension
- 5. Introduction
- 6. Learning to appreciate dimension
- 7. What is dimension?
- 8. Three-dimensional figures
- 9. Physics and dimension