Explorations in Topology
Map Coloring, Surfaces and KnotsBy
- David Gay
This book gives students a rich experience with low-dimensional topology, enhances their geometrical and topological intuition, empowers them with new approaches to solving problems, and provides them with experiences that would help them make sense of a future, more formal topology course. The innovative story-line style of the text models the problems-solving process, presents the development of concepts in a natural way, and through its informality seduces the reader into engagement with the material. The end-of-chapter Investigations give the reader opportunities to work on a variety of open-ended, non-routine problems, and, through a modified Moore method, to make conjectures from which theorems emerge. The students themselves emerge from these experiences owning concepts and results. The end-of-chapter Notes provide historical background to the chapters ideas, introduce standard terminology, and make connections with mainstream mathematics. The final chapter of projects provides opportunities for continued involvement in research beyond the topics of the book.
Upper division, junior/senior mathematics majors and for high school mathematics teachers; individuals who are interested in innovative approaches to the teaching of advanced undergraduate mathematics; mathematicians/mathematics educators interested/specializing in curriculum development.
Hardbound, 352 Pages
Published: November 2006
Imprint: Academic Press
- Preface Chapter 1: Acme Does Maps and Considers Coloring Them Chapter 2: Acme Adds ToursChapter 3: Acme Collects Data from Maps Chapter 4: Acme Collects More Data, Proves a Theorem, and Returns to Coloring Maps Chapter 5: Acmes Solicitor Proves a Theorem: the Four-Color Conjecture Chapter 6: Acme Adds Doughnuts to Its RepertoireChapter 7: Acme Considers the Möbius Strip Chapter 8: Acme Creates New Worlds: Klein Bottles and Other Surfaces Chapter 9: Acme Makes Order Out of Chaos: Surface Sums and Euler Numbers Chapter 10: Acme Classifies Surfaces Chapter 11: Acme Encounters the Fourth Dimension Chapter 12: Acme Colors Maps on Surfaces: Heawoods Estimate Chapter 13: Acme Gets All Tied Up with KnotsChapter 14: Where to Go from Here: ProjectsIndex