A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World)

by Kenji Ueno

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).

Genres: Mathematics

136 pages, Paperback

First published October 1, 2003

Reviews

[Simona · Rating 5 out of 5]

Yes! I am on a strange streak that would be criticized by my past self of just loving the heck out of some books and simply enjoying them unapologetically. What reading this book feels like, with that state of mind, is someone taking you to a journey to an unfamiliar country, showing you all the sites (describing them in the voice of an inspirational speaker), giving you enough time(because it is a book) to ponder all the things they told you, without holding you accountable to you not understanding everything(like textbooks with format Definition, Theorem,Proof,Theorem,Proof left to reader, Corollary, Definition, Theorem,... do).
I have just one complaint: I missed recommended reading: "If you want to know more, this is a good source"

[John Workman · Rating 4 out of 5]

Very interesting and insightful introduction to modern math topics in a way that's both accessible to anyone interested in the subject while also giving me new examples of understanding in subjects I've studied only briefly in school.

[Lucille Nguyen · Rating 5 out of 5]

A good introduction to topology and geometry, with minimal (at most high-school level) mathematical prerequisites.