Algebraic Curves and Riemann Surfaces for Undergraduates

The Theory of the Donut

Anil Nerode, Noam Greenberg (Autoren)

Buch | Softcover

XIV, 450 Seiten

2023 | 1st ed. 2022
Springer International Publishing (Verlag)
978-3-031-11615-5 (ISBN)

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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.

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 Andre 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.

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.

Erscheinungsdatum	18.01.2023
Zusatzinfo	XIV, 450 p. 1 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	791 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	algebraic curves • Complex Analysis • Elliptic Curves • elliptic functions • Manifolds • path integrals • Riemann surface • Undergraduate Textbook • uniformization of elliptic curves
ISBN-10	3-031-11615-1 / 3031116151
ISBN-13	978-3-031-11615-5 / 9783031116155
Zustand	Neuware