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Explorations in Topology

Map Coloring, Surfaces and Knots

Explorations in Topology gives students a rich experience with low-dimensional topology, enhances their geometrical and topological intuition, empowers them with new approache… Read more

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Description

Explorations in Topology 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 chapter’s 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.

Key features

  • Students begin to solve substantial problems right from the start
  • Ideas unfold through the context of a storyline, and students become actively involved
  • The text models the problem-solving process, presents the development of concepts in a natural way, and helps the reader engage with the material

Readership

Upper division, junior/senior mathematics majors and for high school mathematics teachers; mathematicians/mathematics educators interested/specializing in curriculum development.

Table of contents

1. Acme Does Maps and Considers Coloring Them

2. Acme Adds Tours

3. Acme Collects Data from Maps

4. Acme Collects More Data, Proves a Theorem, and Returns to Coloring Maps

5. Acme’s Solicitor Proves a Theorem. the Four-Color Conjecture

6. Acme Adds Doughnuts to Its Repertoire

7. Acme Considers the Möbius Strip

8. Acme Creates New Worlds. Klein Bottles and Other Surfaces

9. Acme Makes Order Out of Chaos. Surface Sums and Euler Numbers

10. Acme Classifies Surfaces

11. Acme Encounters the Fourth Dimension

12. Acme Colors Maps on Surfaces. Heawood’s Estimate

13. Acme Gets All Tied Up with Knots

14. Where to Go from Here. Projects

Product details

About the author

DG

David Gay

Affiliations and expertise
Department of Mathematics, University of Arizona, Tucson, AZ, USA