Introduction to Intersection Theory in Algebraic Geometry

Author: William Fulton

Publisher: American Mathematical Soc.

ISBN: 0821807048

Category: Mathematics

Page: 82

View: 1174

This book introduces some of the main ideas of modern intersection theory, traces their origins in classical geometry and sketches a few typical applications. It requires little technical background: much of the material is accessible to graduate students in mathematics. A broad survey, the book touches on many topics, most importantly introducing a powerful new approach developed by the author and R. MacPherson. It was written from the expository lectures delivered at the NSF-supported CBMS conference at George Mason University, held June 27-July 1, 1983.The author describes the construction and computation of intersection products by means of the geometry of normal cones. In the case of properly intersecting varieties, this yields Samuel's intersection multiplicity; at the other extreme it gives the self-intersection formula in terms of a Chern class of the normal bundle; in general it produces the excess intersection formula of the author and R. MacPherson. Among the applications presented are formulas for degeneracy loci, residual intersections, and multiple point loci; dynamic interpretations of intersection products; Schubert calculus and solutions to enumerative geometry problems; and Riemann-Roch theorems.

An Invitation to Algebraic Geometry

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

Publisher: Springer Science & Business Media

ISBN: 1475744978

Category: Mathematics

Page: 164

View: 4467

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.

Algebraic Geometry

A Concise Dictionary

Author: Elena Rubei

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 3110316234

Category: Mathematics

Page: 239

View: 5184

Algebraic geometry has a complicated, difficult language. This book contains a definition, several references and the statements of the main theorems (without proofs) for every of the most common words in this subject. Some terms of related subjects are included. It helps beginners that know some, but not all, basic facts of algebraic geometry to follow seminars and to read papers. The dictionary form makes it easy and quick to consult.

Polytopes, Rings, and K-Theory

Author: Winfried Bruns,Joseph Gubeladze

Publisher: Springer Science & Business Media

ISBN: 0387763562

Category: Mathematics

Page: 461

View: 1423

This book examines interactions of polyhedral discrete geometry and algebra. What makes this book unique is the presentation of several central results in all three areas of the exposition - from discrete geometry, to commutative algebra, and K-theory.

Hopf Algebras and Their Actions on Rings

Author: Susan Montgomery

Publisher: American Mathematical Soc.

ISBN: 0821807382

Category: Mathematics

Page: 238

View: 3199

The last ten years have seen a number of significant advances in Hopf algebras. The best known is the introduction of quantum groups, which are Hopf algebras that arose in mathematical physics and now have connections to many areas of mathematics. In addition, several conjectures of Kaplansky have been solved, the most striking of which is a kind of Lagrange's theorem for Hopf algebras. Work on actions of Hopf algebras has unified earlier results on group actions, actions of Lie algebras, and graded algebras. This book brings together many of these recent developments from the viewpoint of the algebraic structure of Hopf algebras and their actions and coactions. Quantum groups are treated as an important example, rather than as an end in themselves. The two introductory chapters review definitions and basic facts; otherwise, most of the material has not previously appeared in book form. Providing an accessible introduction to Hopf algebras, this book would make an excellent graduate textbook for a course in Hopf algebras or an introduction to quantum groups.

Deformation Theory

Author: Robin Hartshorne

Publisher: Springer Science & Business Media

ISBN: 1441915966

Category: Mathematics

Page: 234

View: 8265

The basic problem of deformation theory in algebraic geometry involves watching a small deformation of one member of a family of objects, such as varieties, or subschemes in a fixed space, or vector bundles on a fixed scheme. In this new book, Robin Hartshorne studies first what happens over small infinitesimal deformations, and then gradually builds up to more global situations, using methods pioneered by Kodaira and Spencer in the complex analytic case, and adapted and expanded in algebraic geometry by Grothendieck. The author includes numerous exercises, as well as important examples illustrating various aspects of the theory. This text is based on a graduate course taught by the author at the University of California, Berkeley.

Singular Loci of Schubert Varieties

Author: Sara Billey,Venkatramani Lakshmibai

Publisher: Birkhauser


Category: Mathematics

Page: 251

View: 6767

The study of singular loci, an important sub-discipline of Schubert varieties, lies at the crossroads of representation theory, algebraic geometry, and combinatorics. To present a clearer understanding of the subject, the authors have recreated and restructured the various theories and approaches previously available only in journal articles. The book gives a systematic presentation of a wide range of topics and presents new results with sufficient examples, as well as numerous tables -- none of which are found elsewhere in the mathematics literature.

Punctured Torus Groups and 2-Bridge Knot Groups (I)

Author: Hirotaka Akiyoshi,Makoto Sakuma,Masaaki Wada,Yasushi Yamashita

Publisher: Springer


Category: Mathematics

Page: 256

View: 495

Here is the first part of a work that provides a full account of Jorgensen's theory of punctured torus Kleinian groups and its generalization. It offers an elementary and self-contained description of Jorgensen's theory with a complete proof. Through various informative illustrations, readers are naturally led to an intuitive, synthetic grasp of the theory, which clarifies how a very simple fuchsian group evolves into complicated Kleinian groups.

Computer Algebra in Scientific Computing

8th International Workshop, CASC 2005, Kalamata, Greece, September 12-16, 2005, Proceedings

Author: Victor G. Ganzha,Ernst W. Mayr,Evgenii V. Vorozhtsov

Publisher: Springer


Category: Computers

Page: 502

View: 3545

This book constitutes the refereed proceedings of the 8th International Workshop on Computer Algebra in Scientific Computing, CASC 2005, held in Kalamata, Greece in September 2005. The 41 revised full papers presented were carefully reviewed and selected from 75 submissions. The topics addressed in the workshop cover all the basic areas of scientific computing as they benefit from the application of computer algebra methods and software: algebraic methods for nonlinear polynomial equations and inequalities, symbolic-numeric methods for differential and differential-algebraic equations, algorithmic and complexity considerations in computer algebra, algebraic methods in geometric modelling, aspects of computer algebra programming languages, automatic reasoning in algebra and geometry, complexity of algebraic problems, exact and approximate computation, parallel symbolic-numeric computation, Internet accessible symbolic and numeric computation, problem-solving environments, symbolic and numerical computation in systems engineering and modelling, computer algebra in industry, solving problems in the natural sciences, numerical simulation using computer algebra systems, mathematical communication.

Graph Theory and Its Applications to Problems of Society

Author: Fred S. Roberts

Publisher: SIAM

ISBN: 089871026X

Category: Mathematics

Page: 122

View: 6216

Explores modern topics in graph theory and its applications to problems in transportation, genetics, pollution, perturbed ecosystems, urban services, and social inequalities. The author presents both traditional and relatively atypical graph-theoretical topics to best illustrate applications.

Introduction to Tropical Geometry

Author: Diane Maclagan,Bernd Sturmfels

Publisher: American Mathematical Soc.

ISBN: 0821851985

Category: Algebraic geometry -- Special varieties -- Toric varieties, Newton polyhedra

Page: 363

View: 5239

Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of the six chapters concludes with problems that will help the readers to practice their tropical skills, and to gain access to the research literature.