**Author**: Rick Miranda

**Publisher:**American Mathematical Soc.

**ISBN:**0821802682

**Category:**Mathematics

**Page:**390

**View:**354

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# Search Results for: algebraic-curves-and-riemann-surfaces-graduate-studies-in-mathematics

**Author**: Rick Miranda

**Publisher:** American Mathematical Soc.

**ISBN:** 0821802682

**Category:** Mathematics

**Page:** 390

**View:** 354

The book was easy to understand, with many examples. The exercises were well chosen, and served to give further examples and developments of the theory. --William Goldman, University of Maryland 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 center stage. But the main examples come from projective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Duality 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 and cohomology are introduced as a unifying device in the latter chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one semester of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-semester course in complex variables or a year-long course in algebraic geometry.

**Author**: Fedor Bogomolov,Tihomir Petrov

**Publisher:** American Mathematical Soc.

**ISBN:** 9780821883488

**Category:** Mathematics

**Page:** 214

**View:** 3804

Algebraic curves have many special properties that make their study particularly rewarding. As a result, curves provide a natural introduction to algebraic geometry. In this book, the authors also bring out aspects of curves that are unique to them and emphasize connections with algebra. This text covers the essential topics in the geometry of algebraic curves, such as line bundles and vector bundles, the Riemann-Roch Theorem, divisors, coherent sheaves, and zeroth and firstcohomology groups. The authors make a point of using concrete examples and explicit methods to ensure that the style is clear and understandable. Several chapters develop the connections between the geometry of algebraic curves and the algebra of one-dimensional fields. This is an interesting topic that israrely found in introductory texts on algebraic geometry. This book makes an excellent text for a first course for graduate students.

**Author**: Thomas Breuer

**Publisher:** Cambridge University Press

**ISBN:** 9780521798099

**Category:** Mathematics

**Page:** 199

**View:** 3569

Addresses a topic from classical analysis using modern algebraic and computational tools. For graduates and researchers.

**Author**: Emilio Bujalance,Francisco Javier Cirre,José Manuel Gamboa,Grzegorz Gromadzki

**Publisher:** Springer

**ISBN:** 364214828X

**Category:** Mathematics

**Page:** 164

**View:** 3101

This monograph covers symmetries of compact Riemann surfaces. It examines the number of conjugacy classes of symmetries, the numbers of ovals of symmetries and the symmetry types of Riemann surfaces.

**Author**: Gerd Fischer

**Publisher:** Springer-Verlag

**ISBN:** 3322803112

**Category:** Mathematics

**Page:** 177

**View:** 1009

Neben den elementaren Dingen, wie Tangenten, Singularitäten und Wendepunkten werden auch schwierigere Begriffe wie lokale Zweige und Geschlecht behandelt. Höhepunkte sind die klassischen Formeln von Plücker und Clebsch, die Beziehungen zwischen verschiedenen globalen und lokalen Invarianten einer Kurve beschreiben.
*A First Course in Hurwitz Theory*

**Author**: Renzo Cavalieri,Eric Miles

**Publisher:** Cambridge University Press

**ISBN:** 1316798933

**Category:** Mathematics

**Page:** N.A

**View:** 8699

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**: C. Orzech

**Publisher:** CRC Press

**ISBN:** 9780824711597

**Category:** Mathematics

**Page:** 240

**View:** 7754

Plane Algebraic Curves is a classroom-tested textbook for advanced undergraduate and beginning graduate students in mathematics. The book introduces the contemporary notions of algebraic varieties, morphisms of varieties, and adeles to the classical subject of plane curves over algebraically closed fields. By restricting the rigorous development of these notions to a traditional context the book makes its subject accessible without extensive algebraic prerequisites. Once the reader's intuition for plane curves has evolved, there is a discussion of how these objects can be generalized to higher dimensional settings. These features, as well as a proof of the Riemann-Roch Theorem based on a combination of geometric and algebraic considerations, make the book a good foundation for more specialized study in algebraic geometry, commutative algebra, and algebraic function fields. Plane Algebraic Curves is suitable for readers with a variety of backgrounds and interests. The book begins with a chapter outlining prerequisites, and contains informal discussions giving an overview of its material and relating it to non-algebraic topics which would be familiar to the general reader. There is an explanation of why the algebraic genus of a hyperelliptic curve agrees with its geometric genus as a compact Riemann surface, as well as a thorough description of how the classically important elliptic curves can be described in various normal forms. The book concludes with a bibliography which students can incorporate into their further studies. Book jacket.
*Im Herzen einer verborgenen Wirklichkeit*

**Author**: Edward Frenkel

**Publisher:** Springer-Verlag

**ISBN:** 3662434210

**Category:** Mathematics

**Page:** 317

**View:** 6664

**Author**: Wilhelm Schlag

**Publisher:** American Mathematical Society

**ISBN:** 0821898477

**Category:** Mathematics

**Page:** 384

**View:** 2357

Complex analysis is a cornerstone of mathematics, making it an essential element of any area of study in graduate mathematics. Schlag's treatment of the subject emphasizes the intuitive geometric underpinnings of elementary complex analysis that naturally lead to the theory of Riemann surfaces. The book begins with an exposition of the basic theory of holomorphic functions of one complex variable. The first two chapters constitute a fairly rapid, but comprehensive course in complex analysis. The third chapter is devoted to the study of harmonic functions on the disk and the half-plane, with an emphasis on the Dirichlet problem. Starting with the fourth chapter, the theory of Riemann surfaces is developed in some detail and with complete rigor. From the beginning, the geometric aspects are emphasized and classical topics such as elliptic functions and elliptic integrals are presented as illustrations of the abstract theory. The special role of compact Riemann surfaces is explained, and their connection with algebraic equations is established. The book concludes with three chapters devoted to three major results: the Hodge decomposition theorem, the Riemann-Roch theorem, and the uniformization theorem. These chapters present the core technical apparatus of Riemann surface theory at this level. This text is intended as a detailed, yet fast-paced intermediate introduction to those parts of the theory of one complex variable that seem most useful in other areas of mathematics, including geometric group theory, dynamics, algebraic geometry, number theory, and functional analysis. More than seventy figures serve to illustrate concepts and ideas, and the many problems at the end of each chapter give the reader ample opportunity for practice and independent study.

**Author**: Frances Clare Kirwan

**Publisher:** Cambridge University Press

**ISBN:** 9780521423533

**Category:** Mathematics

**Page:** 264

**View:** 7150

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.
*Elemente der Funktionentheorie*

**Author**: Wolfgang Fischer,Ingo Lieb

**Publisher:** Springer-Verlag

**ISBN:** 3834893773

**Category:** Mathematics

**Page:** 214

**View:** 7502

In den Bachelor-Studiengängen der Mathematik steht für die Komplexe Analysis (Funktionentheorie) oft nur eine einsemestrige 2-stündige Vorlesung zur Verfügung. Dieses Buch eignet sich als Grundlage für eine solche Vorlesung im 2. Studienjahr. Mit einer guten thematischen Auswahl, vielen Beispielen und ausführlichen Erläuterungen gibt dieses Buch eine Darstellung der Komplexen Analysis, die genau die Grundlagen und den wesentlichen Kernbestand dieses Gebietes enthält. Das Buch bietet über diese Grundausbildung hinaus weiteres Lehrmaterial als Ergänzung, sodass es auch für eine 3- oder 4 –stündige Vorlesung geeignet ist. Je nach Hörerkreis kann der Stoff unterschiedlich erweitert werden. So wurden für den „Bachelor Lehramt“ die geometrischen Aspekte der Komplexen Analysis besonders herausgearbeitet.

**Author**: Egbert Brieskorn,Horst Knörrer

**Publisher:** N.A

**ISBN:** N.A

**Category:** Mathematics

**Page:** 964

**View:** 5547

**Author**: S. M. Natanzon

**Publisher:** American Mathematical Soc.

**ISBN:** 9780821835944

**Category:** Mathematics

**Page:** 160

**View:** 974

The space of all Riemann surfaces (the so-called moduli space) plays an important role in algebraic geometry and its applications to quantum field theory. The present book is devoted to the study of topological properties of this space and of similar moduli spaces, such as the space of real algebraic curves, the space of mappings, and also superanalogs of all these spaces. The book can be used by researchers and graduate students working in algebraic geometry, topology, and mathematical physics.

**Author**: N.A

**Publisher:** N.A

**ISBN:** 9782856291061

**Category:** Algebraic number theory

**Page:** 109

**View:** 2271

*Twistors, Loop Groups, and Riemann Surfaces*

**Author**: N.J. Hitchin,G. B. Segal,R.S. Ward

**Publisher:** Oxford Graduate Texts in Mathe

**ISBN:** 0199676771

**Category:** Mathematics

**Page:** 136

**View:** 6598

Designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors, this book has its origins in a lecture series given by the internationally renowned authors. Written in an accessible, informal style, it fills a gap in the existing literature.

**Author**: Simon Donaldson

**Publisher:** Oxford University Press

**ISBN:** 0198526393

**Category:** Mathematics

**Page:** 286

**View:** 6969

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**: Andrei Bogatyrev

**Publisher:** Springer Science & Business Media

**ISBN:** 3642256341

**Category:** Mathematics

**Page:** 150

**View:** 2845

The problems of conditional optimization of the uniform (or C-) norm for polynomials and rational functions arise in various branches of science and technology. Their numerical solution is notoriously difficult in case of high degree functions. The book develops the classical Chebyshev's approach which gives analytical representation for the solution in terms of Riemann surfaces. The techniques born in the remote (at the first glance) branches of mathematics such as complex analysis, Riemann surfaces and Teichmüller theory, foliations, braids, topology are applied to approximation problems. The key feature of this book is the usage of beautiful ideas of contemporary mathematics for the solution of applied problems and their effective numerical realization. This is one of the few books where the computational aspects of the higher genus Riemann surfaces are illuminated. Effective work with the moduli spaces of algebraic curves provides wide opportunities for numerical experiments in mathematics and theoretical physics.

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