Hurwitz theory, the study of analytic functions among Riemann surfaces, is a classical field in algebraic geometry. Designed for undergraduate study, this classroom-tested text demonstrates the connections between diverse areas of mathematics and features short essays by guest writers as well as over 100 exercises for the reader.
Hurwitz theory, the study of analytic functions among Riemann surfaces, is a classical field in algebraic geometry. Designed for undergraduate study, this classroom-tested text demonstrates the connections between diverse areas of mathematics and features short essays by guest writers as well as over 100 exercises for the reader.
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.
Inhaltsangabe
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.
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.
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