Mostly Surfaces (original) (raw)

Volume: 60; 2011; 314 pp

MSC: Primary 14; 30; 32; 37; 53; 51

This book presents a number of topics related to surfaces, such as Euclidean, spherical and hyperbolic geometry, the fundamental group, universal covering surfaces, Riemannian manifolds, the Gauss-Bonnet Theorem, and the Riemann mapping theorem. The main idea is to get to some interesting mathematics without too much formality. The book also includes some material only tangentially related to surfaces, such as the Cauchy Rigidity Theorem, the Dehn Dissection Theorem, and the Banach–Tarski Theorem.

The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis.

Undergraduate students interested in geometry and topology of surfaces.

• Chapters

• Chapter 1. Book overview

• Part 1. Surfaces and topology

• Chapter 2. Definition of a surface

• Chapter 3. The gluing construction

• Chapter 4. The fundamental group

• Chapter 5. Examples of fundamental groups

• Chapter 6. Covering spaces and the deck group

• Chapter 7. Existence of universal covers

• Part 2. Surfaces and geometry

• Chapter 8. Euclidean geometry

• Chapter 9. Spherical geometry

• Chapter 10. Hyperbolic geometry

• Chapter 11. Riemannian metrics on surfaces

• Chapter 12. Hyperbolic surfaces

• Part 3. Surfaces and complex analysis

• Chapter 13. A primer on complex analysis

• Chapter 14. Disk and plane rigidity

• Chapter 15. The Schwarz-Christoffel transformation

• Chapter 16. Riemann surfaces and uniformization

• Part 4. Flat cone surfaces

• Chapter 17. Flat cone surfaces

• Chapter 18. Translation surfaces and the Veech group

• Part 5. The totality of surfaces

• Chapter 19. Continued fractions

• Chapter 20. Teichmüller space and moduli space

• Chapter 21. Topology of Teichmüller space

• Part 6. Dessert

• Chapter 22. The Banach–Tarski theorem

• Chapter 23. Dehn’s dissection theorem

• Chapter 24. The Cauchy rigidity theorem

• The book contains a lot of interesting basic and more advanced material which is presented in a nice, intuitive yet rigorous way, and, as such, is perfectly suited as an accompanying text or additional reading for a first course on topology or as a basis for a student seminar.
  Mathematical Reviews

• This is a novel, eclectic, and ambitious collection of geometric and topological topics developed as they relate to surfaces ... a terrific volume. Highly recommended.
  CHOICE

• ...a delightful reading. Schwartz gives a beautiful, careful exposition of some of the most elegant ideas, theorems, and proofs in the theory of surfaces. It's an ideal book for casual reading in spare mathematical moments.
  MAA Reviews

• This highly readable book is an excellent introduction to the theory of surfaces, covering a wide variety of topics with references for further reading. Each chapter contains numerous exercises on the material to get the reader thinking about the subjects covered. There are also many diagrams to aid the reader in understanding the material.
  Alastair Fletcher, Zentralblatt MATH

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