# A Mathematical Gift, I: The interplay between topology, functions, geometry, and algebra

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*Kenji Ueno; Koji Shiga; Shigeyuki Morita*

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.

#### Readership

Advanced high-school students and undergraduates in mathematics.

#### Reviews & Endorsements

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