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, modromy 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 n-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.
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 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.