Elementary Algebraic Geometry

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Author: Klaus Hulek

Publisher: American Mathematical Soc.

ISBN: 0821829521

Category: Mathematics

Page: 213

View: 2348

This book is a true introduction to the basic concepts and techniques of algebraic geometry. The language is purposefully kept on an elementary level, avoiding sheaf theory and cohomology theory. The introduction of new algebraic concepts is always motivated by a discussion of the corresponding geometric ideas. The main point of the book is to illustrate the interplay between abstract theory and specific examples. The book contains numerous problems that illustrate the general theory. The text is suitable for advanced undergraduates and beginning graduate students. It contains sufficient material for a one-semester course. The reader should be familiar with the basic concepts of modern algebra. A course in one complex variable would be helpful, but is not necessary.

Advanced Topics in Linear Algebra

Weaving Matrix Problems through the Weyr Form

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Author: Kevin O'Meara,John Clark,Charles Vinsonhaler

Publisher: Oxford University Press

ISBN: 019987817X

Category: Mathematics

Page: 432

View: 1522

The Weyr matrix canonical form is a largely unknown cousin of the Jordan canonical form. Discovered by Eduard Weyr in 1885, the Weyr form outperforms the Jordan form in a number of mathematical situations, yet it remains somewhat of a mystery, even to many who are skilled in linear algebra. Written in an engaging style, this book presents various advanced topics in linear algebra linked through the Weyr form. Kevin O'Meara, John Clark, and Charles Vinsonhaler develop the Weyr form from scratch and include an algorithm for computing it. A fascinating duality exists between the Weyr form and the Jordan form. Developing an understanding of both forms will allow students and researchers to exploit the mathematical capabilities of each in varying situations. Weaving together ideas and applications from various mathematical disciplines, Advanced Topics in Linear Algebra is much more than a derivation of the Weyr form. It presents novel applications of linear algebra, such as matrix commutativity problems, approximate simultaneous diagonalization, and algebraic geometry, with the latter two having topical connections to phylogenetic invariants in biomathematics and multivariate interpolation. Among the related mathematical disciplines from which the book draws ideas are commutative and noncommutative ring theory, module theory, field theory, topology, and algebraic geometry. Numerous examples and current open problems are included, increasing the book's utility as a graduate text or as a reference for mathematicians and researchers in linear algebra.

Enumerative Geometry and String Theory

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Author: Sheldon Katz

Publisher: American Mathematical Soc.

ISBN: 0821836870

Category: Mathematics

Page: 206

View: 1247

Perhaps the most famous example of how ideas from modern physics have revolutionized mathematics is the way string theory has led to an overhaul of enumerative geometry, an area of mathematics that started in the eighteen hundreds. Century-old problems of enumerating geometric configurations have now been solved using new and deep mathematical techniques inspired by physics!The book begins with an insightful introduction to enumerative geometry. From there, the goal becomes explaining the more advanced elements of enumerative algebraic geometry. Along the way, there are some crash courses on intermediate topics which are essential tools for the student of modern mathematics, such as cohomology and other topics in geometry. The physics content assumes nothing beyond a first undergraduate course. The focus is on explaining the action principle in physics, the idea of string theory, and how these directly lead to questions in geometry. Once these topics are in place, the connection between physics and enumerative geometry is made with the introduction of topological quantum field theory and quantum cohomology.

Birationally Rigid Varieties

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Author: Aleksandr V. Pukhlikov

Publisher: American Mathematical Soc.

ISBN: 0821894765

Category: Mathematics

Page: 367

View: 3337

Birational rigidity is a striking and mysterious phenomenon in higher-dimensional algebraic geometry. It turns out that certain natural families of algebraic varieties (for example, three-dimensional quartics) belong to the same classification type as the

Algebraic Groups and Differential Galois Theory

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Author: Teresa Crespo,Zbigniew Hajto

Publisher: American Mathematical Soc.

ISBN: 082185318X

Category: Mathematics

Page: 225

View: 6496

Differential Galois theory has seen intense research activity during the last decades in several directions: elaboration of more general theories, computational aspects, model theoretic approaches, applications to classical and quantum mechanics as well as to other mathematical areas such as number theory. This book intends to introduce the reader to this subject by presenting Picard-Vessiot theory, i.e. Galois theory of linear differential equations, in a self-contained way. The needed prerequisites from algebraic geometry and algebraic groups are contained in the first two parts of the book. The third part includes Picard-Vessiot extensions, the fundamental theorem of Picard-Vessiot theory, solvability by quadratures, Fuchsian equations, monodromy group and Kovacic's algorithm. Over one hundred exercises will help to assimilate the concepts and to introduce the reader to some topics beyond the scope of this book. This book is suitable for a graduate course in differential Galois theory. The last chapter contains several suggestions for further reading encouraging the reader to enter more deeply into different topics of differential Galois theory or related fields.

Elementary Algebraic Geometry

Second Edition

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Author: Keith Kendig

Publisher: Courier Dover Publications

ISBN: 048680187X

Category: Mathematics

Page: 320

View: 3534

Designed to make learning introductory algebraic geometry as easy as possible, this text is intended for advanced undergraduates and graduate students who have taken a one-year course in algebra and are familiar with complex analysis. This newly updated second edition enhances the original treatment's extensive use of concrete examples and exercises with numerous figures that have been specially redrawn in Adobe Illustrator. An introductory chapter that focuses on examples of curves is followed by a more rigorous and careful look at plane curves. Subsequent chapters explore commutative ring theory and algebraic geometry as well as varieties of arbitrary dimension and some elementary mathematics on curves. Upon finishing the text, students will have a foundation for advancing in several different directions, including toward a further study of complex algebraic or analytic varieties or to the scheme-theoretic treatments of algebraic geometry. 2015 edition.

Differentialgeometrie

Kurven - Flächen - Mannigfaltigkeiten

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Author: Wolfgang Kühnel

Publisher: Springer-Verlag

ISBN: 3834896551

Category: Mathematics

Page: 280

View: 5539

Dieses Buch ist eine Einführung in die Differentialgeometrie. Zunächst geht es um die klassischen Aspekte wie die Geometrie von Kurven und Flächen, bevor dann höherdimensionale Flächen sowie abstrakte Mannigfaltigkeiten betrachtet werden. Die Nahtstelle ist dabei das zentrale Kapitel "Die innere Geometrie von Flächen". Dieses führt den Leser bis hin zu dem berühmten Satz von Gauß-Bonnet, der ein entscheidendes Bindeglied zwischen lokaler und globaler Geometrie darstellt. Die zweite Hälfte des Buches ist der Riemannschen Geometrie gewidmet. Den Abschluss bildet ein Kapitel über "Einstein-Räume", die eine große Bedeutung sowohl in der "Reinen Mathematik" als auch in der Allgemeinen Relativitätstheorie von A. Einstein haben. Es wird großer Wert auf Anschaulichkeit gelegt, was durch zahlreiche Abbildungen unterstützt wird. Im Laufe der Neuauflagen wurde der Text erweitert, neue Aufgaben wurden hinzugefügt und am Ende des Buches wurden zusätzliche Hinweise zur Lösung der Übungsaufgaben ergänzt. Der Text wurde für die fünfte Auflage gründlich durchgesehen und an einigen Stellen verbessert.

Ebene algebraische Kurven

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Author: Gerd Fischer

Publisher: Springer-Verlag

ISBN: 3322803112

Category: Mathematics

Page: 177

View: 1113

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.

Elementare Algebraische Geometrie

Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen

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Author: Klaus Hulek

Publisher: Springer-Verlag

ISBN: 3322802213

Category: Mathematics

Page: 167

View: 8616

Dieses Buch gibt eine Einführung in die Algebraische Geometrie. Ziel ist es, die grundlegenden Begriffe und Techniken der algebraischen Geometrie zusammen mit einer Reihe von Beispielen darzustellen.

Rudiments of Algebraic Geometry

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Author: W.E. Jenner

Publisher: Courier Dover Publications

ISBN: 0486818063

Category: Mathematics

Page: 112

View: 4455

Aimed at advanced undergraduate students of mathematics, this concise text covers the basics of algebraic geometry. Topics include affine spaces, projective spaces, rational curves, algebraic sets with group structure, more. 1963 edition.

Elementary Algebra

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Author: Ron Larson

Publisher: Cengage Learning

ISBN: 0547102275

Category: Mathematics

Page: 720

View: 399

Larson IS student success. ELEMENTARY ALGEBRA owes its success to the hallmark features for which the Larson team is known: learning by example, a straightforward and accessible writing style, emphasis on visualization through the use of graphs to reinforce algebraic and numeric solutions and to interpret data, and comprehensive exercise sets. These pedagogical features are carefully coordinated to ensure that students are better able to make connections between mathematical concepts and understand the content. With a bright, appealing design, the new Fifth Edition builds on the Larson tradition of guided learning by incorporating a comprehensive range of student success materials to help develop students’ proficiency and conceptual understanding of algebra. The text also continues coverage and integration of geometry in examples and exercises. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.

Elementary Algebraic Geometry

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Author: K. Kendig

Publisher: Springer Science & Business Media

ISBN: 1461568994

Category: Mathematics

Page: 309

View: 3480

This book was written to make learning introductory algebraic geometry as easy as possible. It is designed for the general first- and second-year graduate student, as well as for the nonspecialist; the only prerequisites are a one-year course in algebra and a little complex analysis. There are many examples and pictures in the book. One's sense of intuition is largely built up from exposure to concrete examples, and intuition in algebraic geometry is no exception. I have also tried to avoid too much generalization. If one under stands the core of an idea in a concrete setting, later generalizations become much more meaningful. There are exercises at the end of most sections so that the reader can test his understanding of the material. Some are routine, others are more challenging. Occasionally, easily established results used in the text have been made into exercises. And from time to time, proofs of topics not covered in the text are sketched and the reader is asked to fill in the details. Chapter I is of an introductory nature. Some of the geometry of a few specific algebraic curves is worked out, using a tactical approach that might naturally be tried by one not familiar with the general methods intro duced later in the book. Further examples in this chapter suggest other basic properties of curves. In Chapter II, we look at curves more rigorously and carefully.

Plane Algebraic Curves

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Author: Gerd Fischer

Publisher: American Mathematical Soc.

ISBN: 0821821229

Category: Mathematics

Page: 231

View: 1666

This is an excellent introduction to algebraic geometry, which assumes only standard undergraduate mathematical topics: complex analysis, rings and fields, and topology. Reading this book will help establish the geometric intuition that lies behind the more advanced ideas and techniques used in the study of higher-dimensional varieties.

Basic Algebraic Geometry 2

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Author: Igorʹ Rostislavovich Shafarevich

Publisher: Springer Science & Business Media

ISBN: 9783540575542

Category: Mathematics

Page: 269

View: 5643

The second volume of Shafarevich's introductory book on algebraic geometry focuses on schemes, complex algebraic varieties and complex manifolds. As with Volume 1 the author has revised the text and added new material, e.g. a section on real algebraic curves. Although the material is more advanced than in Volume 1 the algebraic apparatus is kept to a minimum making the book accessible to non-specialists. It can be read independently of Volume 1 and is suitable for beginning graduate students in mathematics as well as in theoretical physics.

Adventures in Mathematics

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Author: Martin A. Moskowitz

Publisher: World Scientific

ISBN: 9789812794949

Category: Mathematics

Page: 108

View: 7931

Though elementary in nature, this book deals with fundamental issues in mathematics OCo number, algebra, geometry (both Euclidean and non-Euclidean) and topology. These subjects, on an advanced level, are the same ones with which much of current mathematical research is concerned and were themselves research topics of earlier periods. The material is very suitable both for advanced high school students and for college students interested in elementary mathematics from a higher standpoint. It will also be very useful to high school teachers seeking an overview of their subject matter. Contents: What is a Number?: The Positive Integers; The Integers; The Rational Numbers; The Real Numbers; The Integers Revisited; The Complex Numbers; Polynomials and Other Analogues of the Integers; The Algebra on a Dial; Groups, Finite Fields and Linear Algebra: Introduction to Groups and Their Actions; Applications to Finite Fields and the Theory of Numbers; Linear Algebra; Two- and Three-Dimensional Geometry and Topology: The Euclidean Case; Elliptic and Hyperbolic Geometry; Some Two- and Three-Dimensional Topology. Readership: High school and college teachers and students; college pre-calculus teachers and students; general readers with an interest in mathematics but a limited background."

Commutative Algebra

With a View Toward Algebraic Geometry

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Author: David Eisenbud,Professor David Eisenbud

Publisher: Springer Science & Business Media

ISBN: 9780387942698

Category: Mathematics

Page: 785

View: 6828

Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algebraic geometry. To help beginners, the essential ideals from algebraic geometry are treated from scratch. Appendices on homological algebra, multilinear algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text.

Computational Methods in Commutative Algebra and Algebraic Geometry

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Author: Wolmer Vasconcelos

Publisher: Springer Science & Business Media

ISBN: 9783540213116

Category: Mathematics

Page: 408

View: 4073

This ACM volume deals with tackling problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of commutative algebra and algebraic geometry. The discoveries stem from an interdisciplinary branch of research which has been growing steadily over the past decade. The author covers a wide range, from showing how to obtain deep heuristics in a computation of a ring, a module or a morphism, to developing means of solving nonlinear systems of equations - highlighting the use of advanced techniques to bring down the cost of computation. Although intended for advanced students and researchers with interests both in algebra and computation, many parts may be read by anyone with a basic abstract algebra course.

Elementary Algebra

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Author: Harold R. Jacobs

Publisher: New Leaf Publishing Group

ISBN: 0890519854

Category: Mathematics

Page: 746

View: 6360

Designed for high school students and revised for a new generation of learners! Jacobs Elementary Algebra has come to be highly regarded in the education market. This curriculum provides a full year of mathematics in a clearly written format with guidance for teachers as well as for students who are self-directed. Student textbook includes easy-to-follow instruction and selected answers in the back.Lessons are divided into 17 chapters, covering functions and graphs, integers, rational numbers, exponents, polynomials, factoring, fractions, and more.The course builds a solid foundational understanding and application of key concepts. Also Available: The Elementary Algebra Teacher Guide provides a detailed schedule, tests, and test answer keys as well as additional exercises. The Solutions Manual for Elementary Algebra helps the student with understanding the answers from the book.

Algebra I

Textbook for Students of Mathematics

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Author: Alexey L. Gorodentsev

Publisher: Springer

ISBN: 3319452851

Category: Mathematics

Page: 564

View: 8668

This book is the first volume of an intensive “Russian-style” two-year graduate course in abstract algebra, and introduces readers to the basic algebraic structures – fields, rings, modules, algebras, groups, and categories – and explains the main principles of and methods for working with them. The course covers substantial areas of advanced combinatorics, geometry, linear and multilinear algebra, representation theory, category theory, commutative algebra, Galois theory, and algebraic geometry – topics that are often overlooked in standard undergraduate courses. This textbook is based on courses the author has conducted at the Independent University of Moscow and at the Faculty of Mathematics in the Higher School of Economics. The main content is complemented by a wealth of exercises for class discussion, some of which include comments and hints, as well as problems for independent study.