Algebraic curves and Riemann surfaces for undergraduates : the theory of the donut / Anil Nerode, Noam Greenberg.
Nerode, Anil, 1932- author.; Greenberg, Noam, author.
2022
QA565
Available Online

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Title Algebraic curves and Riemann surfaces for undergraduates : the theory of the donut / Anil Nerode, Noam Greenberg.
Author Nerode, Anil, 1932- author.
ISBN 9783031116162 (electronic bk.)
303111616X (electronic bk.)
3031116151
9783031116155
DOI https://doi.org/10.1007/978-3-031-11616-2
Imprint Cham : Springer, 2022.
Language English
Description 1 online resource
Other Standard Identifiers 10.1007/978-3-031-11616-2 doi
Call Number QA565
Dewey Decimal Classification 516.3/52
Summary 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.
Bibliography, etc. Note Includes bibliographical references and index.
Access Note Access limited to authorized users.
Source of Description Online resource; title from PDF title page (SpringerLink, viewed January 30, 2023).
Added Author Greenberg, Noam, author.
Available in Other Form Print version: 9783031116155
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Record Appears in Online Resources > Ebooks
All Resources
1 Introduction
Part I Algebraic curves
2 Algebra
3 Affine space
4 Projective space
5 Tangents
6 Bezouts 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.

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