Riemann Surfaces and Algebraic Curves

A First Course in Hurwitz Theory

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Author: Renzo Cavalieri,Eric Miles

Publisher: Cambridge University Press

ISBN: 1316798933

Category: Mathematics

Page: N.A

View: 3888

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.

Complex Algebraic Curves

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Author: Frances Clare Kirwan

Publisher: Cambridge University Press

ISBN: 9780521423533

Category: Mathematics

Page: 264

View: 624

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.

Complex Functions

An Algebraic and Geometric Viewpoint

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Author: Gareth A. Jones,David Singerman

Publisher: Cambridge University Press

ISBN: 9780521313667

Category: Mathematics

Page: 342

View: 8121

Elliptic functions and Riemann surfaces played an important role in nineteenth-century mathematics. At the present time there is a great revival of interest in these topics not only for their own sake but also because of their applications to so many areas of mathematical research from group theory and number theory to topology and differential equations. In this book the authors give elementary accounts of many aspects of classical complex function theory including Möbius transformations, elliptic functions, Riemann surfaces, Fuchsian groups and modular functions. A distinctive feature of their presentation is the way in which they have incorporated into the text many interesting topics from other branches of mathematics. This book is based on lectures given to advanced undergraduates and is well-suited as a textbook for a second course in complex function theory. Professionals will also find it valuable as a straightforward introduction to a subject which is finding widespread application throughout mathematics.

Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication

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Author: Christian Rohde

Publisher: Springer

ISBN: 3642006396

Category: Mathematics

Page: 228

View: 2281

Calabi-Yau manifolds have been an object of extensive research during the last two decades. One of the reasons is the importance of Calabi-Yau 3-manifolds in modern physics - notably string theory. An interesting class of Calabi-Yau manifolds is given by those with complex multiplication (CM). Calabi-Yau manifolds with CM are also of interest in theoretical physics, e. g. in connection with mirror symmetry and black hole attractors. It is the main aim of this book to construct families of Calabi-Yau 3-manifolds with dense sets of ?bers with complex multiplication. Most - amples in this book are constructed using families of curves with dense sets of ?bers with CM. The contents of this book can roughly be divided into two parts. The ?rst six chapters deal with families of curves with dense sets of CM ?bers and introduce the necessary theoretical background. This includes among other things several aspects of Hodge theory and Shimura varieties. Using the ?rst part, families of Calabi-Yau 3-manifolds with dense sets of ?bers withCM are constructed in the remaining ?ve chapters. In the appendix one ?nds examples of Calabi-Yau 3-manifolds with complex mul- plication which are not necessarily ?bers of a family with a dense set ofCM ?bers. The author hopes to have succeeded in writing a readable book that can also be used by non-specialists.

Introduction to Plane Algebraic Curves

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Author: Ernst Kunz

Publisher: Springer Science & Business Media

ISBN: 0817644431

Category: Mathematics

Page: 294

View: 5239

* Employs proven conception of teaching topics in commutative algebra through a focus on their applications to algebraic geometry, a significant departure from other works on plane algebraic curves in which the topological-analytic aspects are stressed *Requires only a basic knowledge of algebra, with all necessary algebraic facts collected into several appendices * Studies algebraic curves over an algebraically closed field K and those of prime characteristic, which can be applied to coding theory and cryptography * Covers filtered algebras, the associated graded rings and Rees rings to deduce basic facts about intersection theory of plane curves, applications of which are standard tools of computer algebra * Examples, exercises, figures and suggestions for further study round out this fairly self-contained textbook

Compactness and Contradiction

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Author: Terence Tao

Publisher: American Mathematical Soc.

ISBN: 0821894927

Category: Mathematics

Page: 256

View: 3843

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and nonrigorous to be discussed in the formal literature. Traditionally, it was a matter

Graphs, Surfaces and Homology

An Introduction to Algebraic Topology

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Author: P. Giblin

Publisher: Springer Science & Business Media

ISBN: 9400959532

Category: Science

Page: 329

View: 5902

viii homology groups. A weaker result, sufficient nevertheless for our purposes, is proved in Chapter 5, where the reader will also find some discussion of the need for a more powerful in variance theorem and a summary of the proof of such a theorem. Secondly the emphasis in this book is on low-dimensional examples the graphs and surfaces of the title since it is there that geometrical intuition has its roots. The goal of the book is the investigation in Chapter 9 of the properties of graphs in surfaces; some of the problems studied there are mentioned briefly in the Introduction, which contains an in formal survey of the material of the book. Many of the results of Chapter 9 do indeed generalize to higher dimensions (and the general machinery of simplicial homology theory is avai1able from earlier chapters) but I have confined myself to one example, namely the theorem that non-orientable closed surfaces do not embed in three-dimensional space. One of the principal results of Chapter 9, a version of Lefschetz duality, certainly generalizes, but for an effective presentation such a gener- ization needs cohomology theory. Apart from a brief mention in connexion with Kirchhoff's laws for an electrical network I do not use any cohomology here. Thirdly there are a number of digressions, whose purpose is rather to illuminate the central argument from a slight dis tance, than to contribute materially to its exposition.

Introduction to Compact Riemann Surfaces and Dessins D'Enfants

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Author: Ernesto Girondo,Gabino González-Diez

Publisher: Cambridge University Press

ISBN: 0521519632

Category: Mathematics

Page: 298

View: 9219

An elementary account of the theory of compact Riemann surfaces and an introduction to the Belyi-Grothendieck theory of dessins d'enfants.

Combinatorial Commutative Algebra

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Author: Ezra Miller,Bernd Sturmfels

Publisher: Springer Science & Business Media

ISBN: 0387271031

Category: Mathematics

Page: 420

View: 9173

Recent developments are covered Contains over 100 figures and 250 exercises Includes complete proofs

Geometry of Riemann Surfaces

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Author: Frederick P. Gardiner,Gabino González-Diez,Christos Kourouniotis

Publisher: Cambridge University Press

ISBN: 0521733073

Category: Mathematics

Page: 395

View: 3504

Riemann surfaces is a thriving area of mathematics with applications to hyperbolic geometry, complex analysis, conformal dynamics, discrete groups, algebraic curves and more. This collection of articles presents original research and expert surveys of important related topics, making the field accessible to research workers, graduate students and teachers.

A First Course in Algebraic Topology

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Author: Czes Kosniowski

Publisher: CUP Archive

ISBN: 9780521298643

Category: Mathematics

Page: 269

View: 1212

This self-contained introduction to algebraic topology is suitable for a number of topology courses. It consists of about one quarter 'general topology' (without its usual pathologies) and three quarters 'algebraic topology' (centred around the fundamental group, a readily grasped topic which gives a good idea of what algebraic topology is). The book has emerged from courses given at the University of Newcastle-upon-Tyne to senior undergraduates and beginning postgraduates. It has been written at a level which will enable the reader to use it for self-study as well as a course book. The approach is leisurely and a geometric flavour is evident throughout. The many illustrations and over 350 exercises will prove invaluable as a teaching aid. This account will be welcomed by advanced students of pure mathematics at colleges and universities.

Algebraic Groups

The Theory of Group Schemes of Finite Type over a Field

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Author: J. S. Milne

Publisher: Cambridge University Press

ISBN: 1107167485

Category: Mathematics

Page: 682

View: 5909

Comprehensive introduction to the theory of algebraic group schemes over fields, based on modern algebraic geometry, with few prerequisites.

The Evanston Colloquium

Lectures on Mathematics Delivered from Aug. 28 to Sept. 9, 1893 Before Members of the Congress of Mathematics Held in Connection with the World's Fair in Chicago at Northwestern University, Evanston, Ill

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Author: Felix Klein

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: 109

View: 6464

Automorphisms of Surfaces After Nielsen and Thurston

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Author: Andrew J. Casson,Steven A. Bleiler

Publisher: Cambridge University Press

ISBN: 9780521349857

Category: Mathematics

Page: 104

View: 5014

A comprehensive introduction to selected aspects of modern low-dimensional topology for readers with a knowledge of basic algebra.

Computational Algebraic Geometry

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Author: Hal Schenck

Publisher: Cambridge University Press

ISBN: 9780521536509

Category: Mathematics

Page: 193

View: 5855

This 2003 book investigates interplay between algebra and geometry. Covers: homological algebra, algebraic combinatorics and algebraic topology, and algebraic geometry.

Lectures on K3 Surfaces

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Author: Daniel Huybrechts

Publisher: Cambridge University Press

ISBN: 1316797252

Category: Mathematics

Page: N.A

View: 7975

K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.

An Invitation to Algebraic Geometry

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Author: Karen E. Smith,Lauri Kahanpää,Pekka Kekäläinen,William Traves

Publisher: Springer Science & Business Media

ISBN: 1475744978

Category: Mathematics

Page: 164

View: 7109

This is a description of the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today. It is intended for the working or the aspiring mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites. Few algebraic prerequisites are presumed beyond a basic course in linear algebra.

Lectures on Generating Functions

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Author: Sergei K. Lando

Publisher: American Mathematical Soc.

ISBN: 0821834819

Category: Mathematics

Page: 148

View: 7066

In combinatorics, one often considers the process of enumerating objects of a certain nature, which results in a sequence of positive integers. With each such sequence, one can associate a generating function, whose properties tell us a lot about the nature of the objects being enumerated. Nowadays, the language of generating functions is the main language of enumerative combinatorics. This book is based on the course given by the author at the College of Mathematics of the Independent University of Moscow. It starts with definitions, simple properties, and numerous examples of generating functions. It then discusses various topics, such as formal grammars, generating functions in several variables, partitions and decompositions, and the exclusion-inclusion principle. In the final chapter, the author describes applications of generating functions to enumeration of trees, plane graphs, and graphs embedded in two-dimensional surfaces. Throughout the book, the reader is motivated by interesting examples rather than by general theories. It also contains a lot of exercises to help the reader master the material. Little beyond the standard calculus course is necessary to understand the book. It can serve as a text for a one-semester undergraduate course in combinatorics.

Algebraic Topology

A First Course

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Author: William Fulton

Publisher: Springer Science & Business Media

ISBN: 1461241804

Category: Mathematics

Page: 430

View: 9510

To the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the re lations of these ideas with other areas of mathematics. Rather than choosing one point of view of modem topology (homotopy theory, simplicial complexes, singular theory, axiomatic homology, differ ential topology, etc.), we concentrate our attention on concrete prob lems in low dimensions, introducing only as much algebraic machin ery as necessary for the problems we meet. This makes it possible to see a wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topol ogists-without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical devel opment of the subject. What would we like a student to know after a first course in to pology (assuming we reject the answer: half of what one would like the student to know after a second course in topology)? Our answers to this have guided the choice of material, which includes: under standing the relation between homology and integration, first on plane domains, later on Riemann surfaces and in higher dimensions; wind ing numbers and degrees of mappings, fixed-point theorems; appli cations such as the Jordan curve theorem, invariance of domain; in dices of vector fields and Euler characteristics; fundamental groups

3264 and All That

A Second Course in Algebraic Geometry

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Author: David Eisenbud,Joe Harris

Publisher: Cambridge University Press

ISBN: 1316679381

Category: Mathematics

Page: N.A

View: 9226

This book can form the basis of a second course in algebraic geometry. As motivation, it takes concrete questions from enumerative geometry and intersection theory, and provides intuition and technique, so that the student develops the ability to solve geometric problems. The authors explain key ideas, including rational equivalence, Chow rings, Schubert calculus and Chern classes, and readers will appreciate the abundant examples, many provided as exercises with solutions available online. Intersection is concerned with the enumeration of solutions of systems of polynomial equations in several variables. It has been an active area of mathematics since the work of Leibniz. Chasles' nineteenth-century calculation that there are 3264 smooth conic plane curves tangent to five given general conics was an important landmark, and was the inspiration behind the title of this book. Such computations were motivation for Poincaré's development of topology, and for many subsequent theories, so that intersection theory is now a central topic of modern mathematics.