You are here

A Mathematical Gift I: The Interplay between Topology, Functions, Geometry, and Algebra

Kenji Ueno, Koji Shiga and Shigeyuki Morita
American Mathematical Society
Publication Date: 
Number of Pages: 
Mathematical World 19
[Reviewed by
Hortensia Soto-Johnson
, on

Topology, the study of surfaces, is a topic that students usually don't study until they are mathematics graduate students. If students have an opportunity to study topology at the undergraduate level, the focus is generally point-set topology. With A Mathematical Gift I, there is no reason why every undergraduate student should not be exposed to some topology.

The authors of A Mathematical Gift I: The interplay between topology, functions, geometry, and algebra have written a book about topology that is accessible to even high school students. In fact the book was developed after a series of lectures that were presented to high school students at Kyoto University. The authors provide beautiful illustrations and straightforward explanations of sophisticated ideas. Real world and concrete scenarios are used to introduce definitions such as tangent vectors, critical points and vector fields. The authors implement elementary math concepts to introduce and clarify complex concepts and theorems. For example a detailed explanation of parabolas, hyperbolas, and ellipses leads to the "proof" of the Gauss-Bonnet Theorem. The proofs are not the technical proofs found in mathematics books; instead they are presented as consequences of preceding discussions.

The book has two parts. The first, an invitation to topology, highlights the Euler characteristic, vortices, and curvature of a surface. The second part, the story of dimension, commences with the appreciation of dimension, proceeds with defining dimension and concludes with a discussion of three-dimensional figures such as the sphere, torus and Klein bottle. Cardinality, the Peano Curve and Poincaré's philosophy of dimension, all elements of the second part of the book, are presented via elementary but elegant explanations.

The exercises, which are friendly and non-threatening, make The Mathematics Gift I the perfect choice for anyone who conducts summer workshops for high school students. This book is an ideal supplement for graduate students studying topology for the first time especially since the book does not have to be read in order. It is possible for the reader to solely focus on the more challenging concepts. The book is also excellent for undergraduate independent studies. Any undergraduate student, who completed his or her calculus sequence, would be equipped to master the material in this book on his or her own.

I enjoyed reading the book; it was fun to look at sophisticated concepts through an elementary eye. "The interplay between topology, functions, geometry, and algebra" was instructive and motivating. I was impressed with the author's interjection of student suggestions to lessen ambiguity on a given topic. The ability to depict graduate mathematical concepts for high school students to understand is noteworthy. I am sure anyone who reads this book will be wowed by the detail and clarity presented by the authors. Readers of A Mathematical Gift I: The interplay between topology, functions, geometry, and algebra will want to read A Mathematical Gift II: The interplay between topology, functions, geometry, and algebra.

Hortensia Soto-Johnson is currently Associate Professor of Mathematics at Colorado State University-Pueblo. Her interests include mathematics education and geometry. During her free time she enjoys spending time with her husband Roger, son Miguel and dog Chulitas. She also enjoys reading and doing yoga. She can be reached at

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