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Riemann Surfaces and Algebraic Curves
A First Course in Hurwitz Theory

$50.99 (P)

Part of London Mathematical Society Student Texts

  • Date Published: September 2016
  • availability: In stock
  • format: Paperback
  • isbn: 9781316603529

$ 50.99 (P)

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About the Authors
  • 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.

    • A self-contained reference on Hurwitz theory which brings together material dispersed across the literature
    • Demonstrates connections between complex analysis, algebra, geometry, topology, representation theory and physics
    • Provides everything a geometer needs to offer a course on Hurwitz theory
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    Reviews & endorsements

    '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

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    Product details

    • Date Published: September 2016
    • format: Paperback
    • isbn: 9781316603529
    • length: 194 pages
    • dimensions: 228 x 153 x 13 mm
    • weight: 0.3kg
    • contains: 50 b/w illus. 130 exercises
    • availability: In stock
  • Table of Contents

    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?

  • Authors

    Renzo Cavalieri, Colorado State University
    Renzo Cavalieri is Associate Professor of Mathematics at Colorado State University. He received his PhD in 2005 at the University of Utah under the direction of Aaron Bertram. Hurwitz theory has been an important feature and tool in Cavalieri's research, which revolves around the interaction among moduli spaces of curves and maps from curves, and their different compactifications. He has taught courses on Hurwitz theory at the graduate and undergraduate level at Colorado State University and around the world at the National Institute for Pure and Applied Mathematics (IMPA) in Brazil, Beijing University, and the University of Costa Rica.

    Eric Miles, Colorado Mesa University
    Eric Miles is Assistant Professor of Mathematics at Colorado Mesa University. He received his PhD in 2014 under the supervision of Renzo Cavalieri. Miles' doctoral work was on Bridgeland Stability Conditions, an area of algebraic geometry that makes significant use of homological algebra.

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