Algebraic Curves And Riemann Surfaces For Undergraduates
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Author | : Rick Miranda |
Publisher | : American Mathematical Soc. |
Total Pages | : 414 |
Release | : 1995 |
Genre | : Mathematics |
ISBN | : 0821802682 |
Download Algebraic Curves and Riemann Surfaces Book in PDF, Epub and Kindle
In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking centre stage. But the main examples come fromprojective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Dualtiy Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves andcohomology are introduced as a unifying device in the later chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one term of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-term course in complex variables or a year-long course in algebraic geometry.
Author | : Anil Nerode |
Publisher | : Springer Nature |
Total Pages | : 453 |
Release | : 2023-01-16 |
Genre | : Mathematics |
ISBN | : 303111616X |
Download Algebraic Curves and Riemann Surfaces for Undergraduates Book in PDF, Epub and Kindle
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.
Author | : Rick Miranda |
Publisher | : American Mathematical Soc. |
Total Pages | : 418 |
Release | : 1995 |
Genre | : Mathematics |
ISBN | : 9780821802687 |
Download Algebraic Curves and Riemann Surfaces Book in PDF, Epub and Kindle
In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking centre stage. But the main examples come fromprojective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Dualtiy Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves andcohomology are introduced as a unifying device in the later chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one term of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-term course in complex variables or a year-long course in algebraic geometry.
Author | : Renzo Cavalieri |
Publisher | : Cambridge University Press |
Total Pages | : 197 |
Release | : 2016-09-26 |
Genre | : Mathematics |
ISBN | : 1316798933 |
Download Riemann Surfaces and Algebraic Curves Book in PDF, Epub and Kindle
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, monodromy 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 non-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.
Author | : Kichoon Yang |
Publisher | : World Scientific |
Total Pages | : 184 |
Release | : 1988-11-01 |
Genre | : Mathematics |
ISBN | : 9814520039 |
Download Compact Riemann Surfaces And Algebraic Curves Book in PDF, Epub and Kindle
This volume is an introduction to the theory of Compact Riemann Surfaces and algebraic curves. It gives a concise account of the elementary aspects of different viewpoints in curve theory. Foundational results on divisors and compact Riemann surfaces are also stated and proved.
Author | : Frances Clare Kirwan |
Publisher | : Cambridge University Press |
Total Pages | : 278 |
Release | : 1992-02-20 |
Genre | : Mathematics |
ISBN | : 9780521423533 |
Download Complex Algebraic Curves Book in PDF, Epub and Kindle
This development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics. They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis.
Author | : Simon Donaldson |
Publisher | : Oxford University Press |
Total Pages | : 301 |
Release | : 2011-03-24 |
Genre | : Mathematics |
ISBN | : 0198526393 |
Download Riemann Surfaces Book in PDF, Epub and Kindle
An authoritative but accessible text on one dimensional complex manifolds or Riemann surfaces. Dealing with the main results on Riemann surfaces from a variety of points of view; it pulls together material from global analysis, topology, and algebraic geometry, and covers the essential mathematical methods and tools.
Author | : Otto Forster |
Publisher | : Springer Science & Business Media |
Total Pages | : 262 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461259614 |
Download Lectures on Riemann Surfaces Book in PDF, Epub and Kindle
This book grew out of lectures on Riemann surfaces given by Otto Forster at the universities of Munich, Regensburg, and Münster. It provides a concise modern introduction to this rewarding subject, as well as presenting methods used in the study of complex manifolds in the special case of complex dimension one. From the reviews: "This book deserves very serious consideration as a text for anyone contemplating giving a course on Riemann surfaces."—-MATHEMATICAL REVIEWS
Author | : V.I. Danilov |
Publisher | : Springer Science & Business Media |
Total Pages | : 328 |
Release | : 1998-03-17 |
Genre | : Mathematics |
ISBN | : 9783540637059 |
Download Algebraic Geometry I Book in PDF, Epub and Kindle
"... To sum up, this book helps to learn algebraic geometry in a short time, its concrete style is enjoyable for students and reveals the beauty of mathematics." --Acta Scientiarum Mathematicarum
Author | : Maurizio Cornalba |
Publisher | : World Scientific |
Total Pages | : 716 |
Release | : 1989-06-01 |
Genre | : Mathematics |
ISBN | : 9814590878 |
Download Lectures On Riemann Surfaces - Proceedings Of The College On Riemann Surfaces Book in PDF, Epub and Kindle