Geometry, Algebraic

Foundations of Algebraic Geometry

Author: André Weil

Publisher: American Mathematical Soc.

ISBN: 9780821874622

Category: Geometry, Algebraic

Page: 363

View: 7170

The manuscript of the first edition was completed in 1944. In this revision, the first six chapters of the first edition have been reproduced, and the following chapters rewritten completely. The book is not the last word on the topics it deals with, but the newly written chapters include material which was not to be found except in scattered form, in the literature of the last fifteen years. - Foreword.
Mathematics

Foundations of Algebraic Geometry

Author: AndrŽ Weil

Publisher: American Mathematical Soc.

ISBN: 0821810294

Category: Mathematics

Page: 363

View: 9367

This classic is one of the cornerstones of modern algebraic geometry. At the same time, it is entirely self-contained, assuming no knowledge whatsoever of algebraic geometry, and no knowledge of modern algebra beyond the simplest facts about abstract fields and their extensions, and the bare rudiments of the theory of ideals.
Computers

Foundations of Geometric Algebra Computing

Author: Dietmar Hildenbrand

Publisher: Springer Science & Business Media

ISBN: 3642317944

Category: Computers

Page: 196

View: 2090

The author defines “Geometric Algebra Computing” as the geometrically intuitive development of algorithms using geometric algebra with a focus on their efficient implementation, and the goal of this book is to lay the foundations for the widespread use of geometric algebra as a powerful, intuitive mathematical language for engineering applications in academia and industry. The related technology is driven by the invention of conformal geometric algebra as a 5D extension of the 4D projective geometric algebra and by the recent progress in parallel processing, and with the specific conformal geometric algebra there is a growing community in recent years applying geometric algebra to applications in computer vision, computer graphics, and robotics. This book is organized into three parts: in Part I the author focuses on the mathematical foundations; in Part II he explains the interactive handling of geometric algebra; and in Part III he deals with computing technology for high-performance implementations based on geometric algebra as a domain-specific language in standard programming languages such as C++ and OpenCL. The book is written in a tutorial style and readers should gain experience with the associated freely available software packages and applications. The book is suitable for students, engineers, and researchers in computer science, computational engineering, and mathematics.
Mathematics

Foundations of Algebraic Topology

Author: Samuel Eilenberg,Norman Steenrod

Publisher: Princeton University Press

ISBN: 1400877490

Category: Mathematics

Page: 346

View: 8119

The need for an axiomatic treatment of homology and cohomology theory has long been felt by topologists. Professors Eilenberg and Steenrod present here for the first time an axiomatization of the complete transition from topology to algebra. Originally published in 1952. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Mathematics

Fundamentals of Diophantine Geometry

Author: S. Lang

Publisher: Springer Science & Business Media

ISBN: 9780387908373

Category: Mathematics

Page: 370

View: 2851

Diophantine problems represent some of the strongest aesthetic attractions to algebraic geometry. They consist in giving criteria for the existence of solutions of algebraic equations in rings and fields, and eventually for the number of such solutions. The fundamental ring of interest is the ring of ordinary integers Z, and the fundamental field of interest is the field Q of rational numbers. One discovers rapidly that to have all the technical freedom needed in handling general problems, one must consider rings and fields of finite type over the integers and rationals. Furthermore, one is led to consider also finite fields, p-adic fields (including the real and complex numbers) as representing a localization of the problems under consideration. We shall deal with global problems, all of which will be of a qualitative nature. On the one hand we have curves defined over say the rational numbers. Ifthe curve is affine one may ask for its points in Z, and thanks to Siegel, one can classify all curves which have infinitely many integral points. This problem is treated in Chapter VII. One may ask also for those which have infinitely many rational points, and for this, there is only Mordell's conjecture that if the genus is :;;; 2, then there is only a finite number of rational points.
Mathematics

Methods of Algebraic Geometry

Author: W. V. D. Hodge,Daniel Pedoe

Publisher: Cambridge University Press

ISBN: 9780521469005

Category: Mathematics

Page: 440

View: 7621

This classic work (first published in 1947), in three volumes, provides a lucid and rigorous account of the foundations of modern algebraic geometry. The authors have confined themselves to fundamental concepts and geometrical methods, and do not give detailed developments of geometrical properties but geometrical meaning has been emphasized throughout. This first volume is divided into two parts. The first is devoted to pure algebra: the basic notions, the theory of matrices over a non-commutative ground field and a study of algebraic equations. The second part is in n dimensions. It concludes with a purely algebraic account of collineations and correlations.
Science

Algebraic Foundations of Non-Commutative Differential Geometry and Quantum Groups

Author: Ludwig Pittner

Publisher: Springer Science & Business Media

ISBN: 3540478019

Category: Science

Page: 469

View: 2460

Quantum groups and quantum algebras as well as non-commutative differential geometry are important in mathematics and considered to be useful tools for model building in statistical and quantum physics. This book, addressing scientists and postgraduates, contains a detailed and rather complete presentation of the algebraic framework. Introductory chapters deal with background material such as Lie and Hopf superalgebras, Lie super-bialgebras, or formal power series. Great care was taken to present a reliable collection of formulae and to unify the notation, making this volume a useful work of reference for mathematicians and mathematical physicists.
Mathematics

Algebraical and Topological Foundations of Geometry

Proceedings of a Colloquium Held in Utrecht, August 1959

Author: Hans Freudenthal

Publisher: Elsevier

ISBN: 1483184641

Category: Mathematics

Page: 216

View: 1398

Algebraical and Topological Foundations of Geometry contains the proceedings of the Colloquium on Algebraic and Topological Foundations of Geometry, held in Utrecht, the Netherlands in August 1959. The papers review the algebraical and topological foundations of geometry and cover topics ranging from the geometric algebra of the Möbius plane to the theory of parallels with applications to closed geodesies. Groups of homeomorphisms and topological descriptive planes are also discussed. Comprised of 26 chapters, this book introduces the reader to the theory of parallels with applications to closed geodesies; groups of homeomorphisms; complemented modular lattices; and topological descriptive planes. Subsequent chapters focus on collineation groups; exceptional algebras and exceptional groups; the connection between algebra and constructions with ruler and compasses; and the use of differential geometry and analytic group theory methods in foundations of geometry. Von Staudt projectivities of Moufang planes are also considered, and an axiomatic treatment of polar geometry is presented. This monograph will be of interest to students of mathematics.
Mathematics

Algebraic Geometry

Author: Robin Hartshorne

Publisher: Springer Science & Business Media

ISBN: 1475738498

Category: Mathematics

Page: 496

View: 2270

An introduction to abstract algebraic geometry, with the only prerequisites being results from commutative algebra, which are stated as needed, and some elementary topology. More than 400 exercises distributed throughout the book offer specific examples as well as more specialised topics not treated in the main text, while three appendices present brief accounts of some areas of current research. This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra. Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. He is the author of "Residues and Duality", "Foundations of Projective Geometry", "Ample Subvarieties of Algebraic Varieties", and numerous research titles.
Mathematics

Hilbert

Author: Constance Reid,Hermann Weyl

Publisher: Springer-Verlag

ISBN: 3662286157

Category: Mathematics

Page: 290

View: 2449

Mathematics

Introduction to Algebraic Geometry

Author: W. Gordon Welchman

Publisher: Cambridge University Press

ISBN: 1316601803

Category: Mathematics

Page: 362

View: 7637

Originally published in 1950, this textbook studies projective geometry and provides a solid introduction to similar studies in space of more than two dimensions.
Mathematics

The Legacy of Mario Pieri in Geometry and Arithmetic

Author: Elena Anne Marchisotto,James T. Smith

Publisher: Springer Science & Business Media

ISBN: 9780817646035

Category: Mathematics

Page: 494

View: 3562

This book is the first in a series of three volumes that comprehensively examine Mario Pieri’s life, mathematical work and influence. The book introduces readers to Pieri’s career and his studies in foundations, from both historical and modern viewpoints. Included in this volume are the first English translations, along with analyses, of two of his most important axiomatizations — one in arithmetic and one in geometry. The book combines an engaging exposition, little-known historical notes, exhaustive references and an excellent index. And yet the book requires no specialized experience in mathematical logic or the foundations of geometry.
Mathematics

Moderne Algebra

Author: Bartel Eckmann L. Van der van der Waerden,Emil Artin,Emmy Noether

Publisher: Springer-Verlag

ISBN: 3662364344

Category: Mathematics

Page: 274

View: 557

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.
Mathematics

Algebraic Geometry I

Algebraic Curves, Algebraic Manifolds and Schemes

Author: V.I. Danilov,V.V. Shokurov

Publisher: Springer Science & Business Media

ISBN: 9783540519959

Category: Mathematics

Page: 310

View: 9981

"... 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
Mathematics

Algebraic Geometry in Coding Theory and Cryptography

Author: Harald Niederreiter,Chaoping Xing

Publisher: Princeton University Press

ISBN: 9781400831302

Category: Mathematics

Page: 272

View: 4119

This textbook equips graduate students and advanced undergraduates with the necessary theoretical tools for applying algebraic geometry to information theory, and it covers primary applications in coding theory and cryptography. Harald Niederreiter and Chaoping Xing provide the first detailed discussion of the interplay between nonsingular projective curves and algebraic function fields over finite fields. This interplay is fundamental to research in the field today, yet until now no other textbook has featured complete proofs of it. Niederreiter and Xing cover classical applications like algebraic-geometry codes and elliptic-curve cryptosystems as well as material not treated by other books, including function-field codes, digital nets, code-based public-key cryptosystems, and frameproof codes. Combining a systematic development of theory with a broad selection of real-world applications, this is the most comprehensive yet accessible introduction to the field available. Introduces graduate students and advanced undergraduates to the foundations of algebraic geometry for applications to information theory Provides the first detailed discussion of the interplay between projective curves and algebraic function fields over finite fields Includes applications to coding theory and cryptography Covers the latest advances in algebraic-geometry codes Features applications to cryptography not treated in other books
Mathematics

Homotopical Algebraic Geometry II: Geometric Stacks and Applications

Geometric Stacks and Applications

Author: Bertrand Toen,Bertrand Toën,Gabriele Vezzosi

Publisher: American Mathematical Soc.

ISBN: 0821840991

Category: Mathematics

Page: 224

View: 3084

This is the second part of a series of papers called ""HAG"", devoted to developing the foundations of homotopical algebraic geometry. The authors start by defining and studying generalizations of standard notions of linear algebra in an abstract monoidal model category, such as derivations, etale and smooth morphisms, flat and projective modules, etc. They then use their theory of stacks over model categories to define a general notion of geometric stack over a base symmetric monoidal model category $C$, and prove that this notion satisfies the expected properties.
Mathematics

Elementary Algebraic Geometry

Second Edition

Author: Keith Kendig

Publisher: Courier Dover Publications

ISBN: 048680187X

Category: Mathematics

Page: 320

View: 7226

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.