Riemann Surfaces and Algebraic Curves
A First Course in Hurwitz Theory

Classroom-tested and featuring over 100 exercises, this text introduces the key algebraic geometry field of Hurwitz theory.

Renzo Cavalieri (Author), Eric Miles (Author)

9781316603529, Cambridge University Press

Paperback / softback, published 26 September 2016

194 pages, 50 b/w illus. 130 exercises
22.8 x 15.3 x 1.3 cm, 0.3 kg

'To wit, the book is indeed well-suited to advanced undergraduates who know some serious algebra, analysis (complex analysis in particular), and are disposed to hit themes in algebraic topology and (to a limited degree) algebraic geometry. It would make a good text for a senior seminar.' Michael Berg, MAA Reviews

Hurwitz theory, the study of analytic functions among Riemann surfaces, is a classical field and active research area in algebraic geometry. The subject's interplay between algebra, geometry, topology and analysis is a beautiful example of the interconnectedness of mathematics. This book introduces students to this increasingly important field, covering key topics such as manifolds, monodromy representations and the Hurwitz potential. Designed for undergraduate study, this classroom-tested text includes over 100 exercises to provide motivation for the reader. Also included are short essays by guest writers on how they use Hurwitz theory in their work, which ranges from string theory to non-Archimedean geometry. Whether used in a course or as a self-contained reference for graduate students, this book will provide an exciting glimpse at mathematics beyond the standard university classes.

Introduction
1. From complex analysis to Riemann surfaces
2. Introduction to manifolds
3. Riemann surfaces
4. Maps of Riemann surfaces
5. Loops and lifts
6. Counting maps
7. Counting monodromy representations
8. Representation theory of Sd
9. Hurwitz numbers and Z(Sd)
10. The Hurwitz potential
Appendix A. Hurwitz theory in positive characteristic
Appendix B. Tropical Hurwitz numbers
Appendix C. Hurwitz spaces
Appendix D. Does physics have anything to say about Hurwitz numbers?
References
Index.

Subject Areas: Topology [PBP], Geometry [PBM], Algebra [PBF]