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Cover: 9783031116155 | Algebraic Curves and Riemann Surfaces for Undergraduates | Taschenbuch
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Algebraic Curves and Riemann Surfaces for Undergraduates
The Theory of the Donut
Taschenbuch von Noam Greenberg (u. a.)
Sprache: Englisch

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Beschreibung
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.
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.
Über den Autor
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.
Zusammenfassung

Assumes only typical undergraduate background in algebra and calculus

Carefully works out the necessary algebra, topology, complex analysis and differential geometry

Includes over 550 exercises

Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2023
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Inhalt: xiv
450 S.
1 s/w Illustr.
450 p. 1 illus.
ISBN-13: 9783031116155
ISBN-10: 3031116151
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Greenberg, Noam
Nerode, Anil
Auflage: 1st ed. 2022
Hersteller: Springer International Publishing
Springer International Publishing AG
Maße: 235 x 155 x 24 mm
Von/Mit: Noam Greenberg (u. a.)
Erscheinungsdatum: 17.01.2023
Gewicht: 0,782 kg
Artikel-ID: 122017134
Über den Autor
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.
Zusammenfassung

Assumes only typical undergraduate background in algebra and calculus

Carefully works out the necessary algebra, topology, complex analysis and differential geometry

Includes over 550 exercises

Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2023
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Inhalt: xiv
450 S.
1 s/w Illustr.
450 p. 1 illus.
ISBN-13: 9783031116155
ISBN-10: 3031116151
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Greenberg, Noam
Nerode, Anil
Auflage: 1st ed. 2022
Hersteller: Springer International Publishing
Springer International Publishing AG
Maße: 235 x 155 x 24 mm
Von/Mit: Noam Greenberg (u. a.)
Erscheinungsdatum: 17.01.2023
Gewicht: 0,782 kg
Artikel-ID: 122017134
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