Geometric Methods in the Algebraic Theory of Quadratic Forms

by Oleg T. Izhboldin
Geometric Methods in the Algebraic Theory of Quadratic Forms
Author: Oleg T. Izhboldin
Publisher: Springer
Total Pages: 198
Release: 2004-02-07
Genre: Mathematics
ISBN: 3540409904
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The geometric approach to the algebraic theory of quadratic forms is the study of projective quadrics over arbitrary fields. Function fields of quadrics have been central to the proofs of fundamental results since the 1960's. Recently, more refined geometric tools have been brought to bear on this topic, such as Chow groups and motives, and have produced remarkable advances on a number of outstanding problems. Several aspects of these new methods are addressed in this volume, which includes an introduction to motives of quadrics by A. Vishik, with various applications, notably to the splitting patterns of quadratic forms, papers by O. Izhboldin and N. Karpenko on Chow groups of quadrics and their stable birational equivalence, with application to the construction of fields with u-invariant 9, and a contribution in French by B. Kahn which lays out a general framework for the computation of the unramified cohomology groups of quadrics and other cellular varieties.

The Ricci Flow in Riemannian Geometry

by Ben Andrews
The Ricci Flow in Riemannian Geometry
Author: Ben Andrews
Publisher: Springer Science & Business Media
Total Pages: 306
Release: 2011
Genre: Mathematics
ISBN: 3642162851
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This book focuses on Hamilton's Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for Riemannian manifolds, and Perelman's noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of Böhm and Wilking and Brendle and Schoen have led to a proof of the differentiable 1/4-pinching sphere theorem.

The Algebraic and Geometric Theory of Quadratic Forms

by Richard S. Elman
The Algebraic and Geometric Theory of Quadratic Forms
Author: Richard S. Elman
Publisher: American Mathematical Soc.
Total Pages: 456
Release: 2008-07-15
Genre: Mathematics
ISBN: 9780821873229
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This book is a comprehensive study of the algebraic theory of quadratic forms, from classical theory to recent developments, including results and proofs that have never been published. The book is written from the viewpoint of algebraic geometry and includes the theory of quadratic forms over fields of characteristic two, with proofs that are characteristic independent whenever possible. For some results both classical and geometric proofs are given. Part I includes classical algebraic theory of quadratic and bilinear forms and answers many questions that have been raised in the early stages of the development of the theory. Assuming only a basic course in algebraic geometry, Part II presents the necessary additional topics from algebraic geometry including the theory of Chow groups, Chow motives, and Steenrod operations. These topics are used in Part III to develop a modern geometric theory of quadratic forms.

Algebraic Theory of Quadratic Numbers

by Mak Trifković
Algebraic Theory of Quadratic Numbers
Author: Mak Trifković
Publisher: Springer Science & Business Media
Total Pages: 206
Release: 2013-09-14
Genre: Mathematics
ISBN: 1461477174
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By focusing on quadratic numbers, this advanced undergraduate or master’s level textbook on algebraic number theory is accessible even to students who have yet to learn Galois theory. The techniques of elementary arithmetic, ring theory and linear algebra are shown working together to prove important theorems, such as the unique factorization of ideals and the finiteness of the ideal class group. The book concludes with two topics particular to quadratic fields: continued fractions and quadratic forms. The treatment of quadratic forms is somewhat more advanced than usual, with an emphasis on their connection with ideal classes and a discussion of Bhargava cubes. The numerous exercises in the text offer the reader hands-on computational experience with elements and ideals in quadratic number fields. The reader is also asked to fill in the details of proofs and develop extra topics, like the theory of orders. Prerequisites include elementary number theory and a basic familiarity with ring theory.

Classical Algebraic Geometry

by Igor V. Dolgachev
Classical Algebraic Geometry
Author: Igor V. Dolgachev
Publisher: Cambridge University Press
Total Pages: 653
Release: 2012-08-16
Genre: Mathematics
ISBN: 1139560786
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Algebraic geometry has benefited enormously from the powerful general machinery developed in the latter half of the twentieth century. The cost has been that much of the research of previous generations is in a language unintelligible to modern workers, in particular, the rich legacy of classical algebraic geometry, such as plane algebraic curves of low degree, special algebraic surfaces, theta functions, Cremona transformations, the theory of apolarity and the geometry of lines in projective spaces. The author's contemporary approach makes this legacy accessible to modern algebraic geometers and to others who are interested in applying classical results. The vast bibliography of over 600 references is complemented by an array of exercises that extend or exemplify results given in the book.