Richard Elman, University of California, Los Angeles, CA, Nikita Karpenko, Universite Pierre et Marie Curie - Paris 6, France, and Alexander Merkurjev, University of California, Los Angeles, CA

The Algebraic and Geometric Theory of Quadratic Forms

Colloquium Publications, Volume: 56
2008; approx. 431 pp; hardcover
ISBN-13: 978-0-8218-4329-1
Expected publication date is March 8, 2008.

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.

Readership

Graduate students and research mathematicians interested in algebraic geometry and number theory.

Table of Contents

Classical theory of symmetric bilinear forms and quadratic forms

Bilinear forms
Quadratic forms
Forms over rational function fields
Function fields of quadrics
Bilinear and quadratic forms and algebraic extensions
u-invariants
Applications of the Milnor conjecture
On the norm residue homomorphism of degree two

Algebraic cycles

Homology and cohomology
Chow groups
Steenrod operations
Category of Chow motives

Quadratic forms and algebraic cycles

Cycles on powers of quadrics
The Izhboldin dimension
Application of Steenrod operations
The variety of maximal totally isotropic subspaces
Motives of quadrics
Appendices
Bibliography
Notation
Terminology

Edited by: David R. Larson, Texas A&M University, College Station, TX, Peter Massopust, Technische Universitat Munchen, Munich, Germany, Zuhair Nashed, University of Central Florida, Orlando, FL, Minh Chuong Nguyen, Vietnamese Academy of Science and Technology, Hanoi, Vietnam, Manos Papadakis, University of Houston, TX, and Ahmed Zayed, DePaul University, Chicago, IL

Frames and Operator Theory in Analysis and Signal Processing

Contemporary Mathematics, Volume: 451
2008; 291 pp; softcover
ISBN-13: 978-0-8218-4144-0
Expected publication date is March 16, 2008.

This volume contains articles based on talks presented at the Special Session Frames and Operator Theory in Analysis and Signal Processing, held in San Antonio, Texas, in January of 2006.

Recently, the field of frames has undergone tremendous advancement. Most of the work in this field is focused on the design and construction of more versatile frames and frames tailored towards specific applications, e.g., finite dimensional uniform frames for cellular communication. In addition, frames are now becoming a hot topic in mathematical research as a part of many engineering applications, e.g., matching pursuits and greedy algorithms for image and signal processing. Topics covered in this book include:

Application of several branches of analysis (e.g., PDEs; Fourier, wavelet, and harmonic analysis; transform techniques; data representations) to industrial and engineering problems, specifically image and signal processing.
Theoretical and applied aspects of frames and wavelets.
Pure aspects of operator theory emphasizing the connections to applied mathematics, frames, and signal processing.
This volume will be equally attractive to pure mathematicians, working on the foundations of frame and operator theory and their interconnections, as it will to applied mathematicians investigating applications, and to physicists and engineers employing these designs. It thus may appeal to a wide target group of researchers and may serve as a catalyst for cross-fertilization of several important areas of mathematics and the applied sciences.

Readership

Graduate students and research mathematicians interested in pure and applied harmonic analysis, and operator theory.

Table of Contents

Edited by: Jacob E. Goodman and Janos Pach, City College, CUNY, New York, NY,
and Richard Pollack, Courant Institute, New York, NY

Surveys on Discrete and Computational Geometry: Twenty Years Later

Contemporary Mathematics, Volume: 453
2008; 556 pp; softcover
ISBN-13: 978-0-8218-4239-3
Expected publication date is March 29, 2008.

This volume contains nineteen survey papers describing the state of current research in discrete and computational geometry as well as a set of open problems presented at the 2006 AMS-IMS-SIAM Summer Research Conference "Discrete and Computational Geometry--Twenty Years Later", held in Snowbird, Utah, in June 2006. Topics surveyed include metric graph theory, lattice polytopes, the combinatorial complexity of unions of geometric objects, line and pseudoline arrangements, algorithmic semialgebraic geometry, persistent homology, unfolding polyhedra, pseudo-triangulations, nonlinear computational geometry, k-sets, and the computational complexity of convex bodies.

Discrete and computational geometry originated as a discipline in the mid-1980s when mathematicians in the well-established field of discrete geometry and computer scientists in the (then) nascent field of computational geometry began working together on problems of common interest. The combined field has experienced a huge growth in the past twenty years, which the present volume attests to.

Readership

Graduate students and research mathematicians interested in discrete and computational geometry.

Table of Contents

A. D. Elmendorf, I. Kriz, M. A. Mandell, J. P. May, and M. Cole

Rings, Modules, and Algebras in Stable Homotopy Theory

Mathematical Surveys and Monographs, Volume: 47
1997; 249 pp; softcover
ISBN-13: 978-0-8218-4303-1
Expected publication date is April 30, 2008.

This book introduces a new point-set level approach to stable homotopy theory that has already had many applications and promises to have a lasting impact on the subject. Given the sphere spectrum S, the authors construct an associative, commutative, and unital smash product in a complete and cocomplete category of "S-modules" whose derived category is equivalent to the classical stable homotopy category. This construction allows for a simple and algebraically manageable definition of "S-algebras" and "commutative S-algebras" in terms of associative, or associative and commutative, products R\wedge _SR \longrightarrow R. These notions are essentially equivalent to the earlier notions of A_{\infty } and E_{\infty } ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of R-modules in terms of maps R\wedge _SM\longrightarrow M. When R is commutative, the category of R-modules also has an associative, commutative, and unital smash product, and its derived category has properties just like the stable homotopy category. These constructions allow the importation into stable homotopy theory of a great deal of point-set level algebra.

Readership

Graduate students and research mathematicians interested in algebraic topology.

Reviews

"Very well organized ... The exposition is quite clear, with just the right amount of motivational comments. All algebraic topologists should obtain some familiarity with the contents of this book."

-- Mathematical Reviews

Table of Contents

Introduction
Prologue: the category of {\mathbb L}-spectra
Structured ring and module spectra
The homotopy theory of R-modules
The algebraic theory of R-modules
R-ring spectra and the specialization to MU
Algebraic K-theory of S-algebras
R-algebras and topological model categories
Bousfield localizations of R-modules and algebras
Topological Hochschild homology and cohomology
Some basic constructions on spectra
Spaces of linear isometries and technical theorems
The monadic bar construction
Epilogue: The category of {\mathbb L}-spectra under S
Appendix A. Twisted half-smash products and function spectra
Bibliography
Index

Edited by: Rita A. Hibschweiler, University of New Hampshire, Durham, NH,
and Thomas H. MacGregor, State University of New York at Albany, NY

Banach Spaces of Analytic Functions

Contemporary Mathematics, Volume: 454
2008; 147 pp; softcover
ISBN-13: 978-0-8218-4268-3
Expected publication date is April 13, 2008.

This volume is focused on Banach spaces of functions analytic in the open unit disc, such as the classical Hardy and Bergman spaces, and weighted versions of these spaces. Other spaces under consideration here include the Bloch space, the families of Cauchy transforms and fractional Cauchy transforms, BMO, VMO, and the Fock space. Some of the work deals with questions about functions in several complex variables.

Multiplication operators, composition operators and weighted composition operators form a central topic of the volume. This topic has been an extensive area of research for the past twenty years. This volume includes results characterizing bounded, compact and isometric composition operators in various settings.

Graduate students who are interested in analysis will find an overview of current work in the field. Specialists will find interesting questions and new methods, as well as familiar ideas (such as composition operators) seen in new settings or in more general form. Mathematicians with an interest in modern analysis will gain insight into the interplay between function theory and operator theory which is central to this work.

Readership

Graduate students and research mathematicians interested in function theory and operator theory.

Table of Contents

Giuseppe Zampieri, University of Padova, Italy

Complex Analysis and CR Geometry

University Lecture Series, Volume: 43
2008; approx. 204 pp; softcover
ISBN-13: 978-0-8218-4442-7
Expected publication date is May 17, 2008.

Cauchy-Riemann (CR) geometry is the study of manifolds equipped with a system of CR-type equations. Compared to the early days when the purpose of CR geometry was to supply tools for the analysis of the existence and regularity of solutions to the \bar\partial-Neumann problem, it has rapidly acquired a life of its own and has became an important topic in differential geometry and the study of non-linear partial differential equations. A full understanding of modern CR geometry requires knowledge of various topics such as real/complex differential and symplectic geometry, foliation theory, the geometric theory of PDE's, and microlocal analysis. Nowadays, the subject of CR geometry is very rich in results, and the amount of material required to reach competence is daunting to graduate students who wish to learn it.

However, the present book does not aim at introducing all the topics of current interest in CR geometry. Instead, an attempt is made to be friendly to the novice by moving, in a fairly relaxed way, from the elements of the theory of holomorphic functions in several complex variables to advanced topics such as extendability of CR functions, analytic discs, their infinitesimal deformations, and their lifts to the cotangent space. The choice of topics provides a good balance between a first exposure to CR geometry and subjects representing current research. Even a seasoned mathematician who wants to contribute to the subject of CR analysis and geometry will find the choice of topics attractive.

Readership

Graduate students and research mathematicians interested in complex analysis and differential geometry.

Table of Contents

Several complex variables
Real structures
Real/complex structures
Bibliography
Subject index
Symbols index