Algebraic Curves and Riemann Surfaces for Undergraduates
The Theory of the Donut
Published on 17. January 2023
Book
Paperback/Softback
XIV, 450 pages
978-3-031-11615-5 (ISBN)
Description
The theory relating algebraic curves and Riemann surfaces exhibits the unity of mathematics: topology, complex analysis, algebra and geometry all interact in a deep way. This textbook offers an elementary introduction to this beautiful theory for an undergraduate audience. At the heart of the subject is the theory of elliptic functions and elliptic curves. A complex torus (or "donut") is both an abelian group and a Riemann surface. It is obtained by identifying points on the complex plane. At the same time, it can be viewed as a complex algebraic curve, with addition of points given by a geometric "chord-and-tangent" method. This book carefully develops all of the tools necessary to make sense of this isomorphism. The exposition is kept as elementary as possible and frequently draws on familiar notions in calculus and algebra to motivate new concepts. Based on a capstone course given to senior undergraduates, this book is intended as a textbook for courses at this level and includes a large number of class-tested exercises. The prerequisites for using the book are familiarity with abstract algebra, calculus and analysis, as covered in standard undergraduate courses.
Reviews / Votes
"The book is very well written, with a number of exercises ... . As a text for a high-level undergraduate course, it is an excellent piece of work." (José Javier Etayo, Mathematical Reviews, August, 2025)
More details
Edition
1st ed. 2022
Language
English
Place of publication
Cham
Switzerland
Publishing group
Springer International Publishing
Illustrations
1 s/w Abbildung
XIV, 450 p. 1 illus.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 24 mm
Weight
782 gr
ISBN-13
978-3-031-11615-5 (9783031116155)
DOI
10.1007/978-3-031-11616-2
Schweitzer Classification
Other editions
Additional editions
Anil Nerode | Noam Greenberg
Algebraic Curves and Riemann Surfaces for Undergraduates
The Theory of the Donut
E-Book
01/2023
Springer
€58.84
Available for download
Persons
Anil Nerode
, Distinguished Professor of Mathematics at Cornell University, has, over a period of 66 years, made significant contributions to mathematical logic, automata theory, computability theory, and hybrid systems engineering, publishing around 100 papers and 5 books. He first learned elliptic function theory as a graduate student from André Weil in the early 1950s and has taught it over the years, resulting in the present book.
Noam Greenberg is Professor of Mathematics at Victoria University of Wellington, New Zealand. His main research interests are computability theory, algorithmic randomness, reverse mathematics, higher recursion theory, computable model theory, and set theory. He was a Royal Society of New Zealand Rutherford Discovery Fellow and is a Fellow of the Royal Society of New Zealand.
Noam Greenberg is Professor of Mathematics at Victoria University of Wellington, New Zealand. His main research interests are computability theory, algorithmic randomness, reverse mathematics, higher recursion theory, computable model theory, and set theory. He was a Royal Society of New Zealand Rutherford Discovery Fellow and is a Fellow of the Royal Society of New Zealand.
Content
1 Introduction.- Part I Algebraic curves.- 2 Algebra.- 3 Affine space.- 4 Projective space.- 5 Tangents.- 6 Bézout's theorem.- 7 The elliptic group.- Part II Riemann Surfaces.- 8 Quasi-Euclidean spaces.- 9 Connectedness, smooth and simple.- 10 Path integrals.- 11 Complex differentiation.- 12 Riemann surfaces.- Part III Curves and surfaces.- 13 Curves are surfaces.- 14 Elliptic functions and the isomorphism theorem.- 15 Puiseux theory.- 16 A brief history of elliptic functions.