Translations of Mathematical Monographs, Volume: 235
Iwanami Series in Modern Mathematics
2007; approx. 224 pp; softcover
ISBN-10: 0-8218-2097-4
ISBN-13: 978-0-8218-2097-1
Expected publication date is December 29, 2007.
The Atiyah-Singer index theorem is a remarkable result that allows one to compute the space of solutions of a linear elliptic partial differential operator on a manifold in terms of purely topological data related to the manifold and the symbol of the operator. First proved by Atiyah and Singer in 1963, it marked the beginning of a completely new direction of research in mathematics with relations to differential geometry, partial differential equations, differential topology, K-theory, physics, and other areas.
The author's main goal in this volume is to give a complete proof of the index theorem. The version of the proof he chooses to present is the one based on the localization theorem. The prerequisites include a first course in differential geometry, some linear algebra, and some facts about partial differential equations in Euclidean spaces.
Readership
Graduate students interested in index theory.
Table of Contents
Prelude
Manifolds, vector bundles and elliptic complexes
Index and its localization
Examples of the localization of the index
Localization of eigenfunctions of the operator of Laplace type
Formulation and proof of the index theorem
Characteristic classes
Index
Student Mathematical Library, Volume: 40
2007; approx. 302 pp; softcover
ISBN-10: 0-8218-4212-9
ISBN-13: 978-0-8218-4212-6
Expected publication date is January 6, 2008.
Frames for Undergraduates is an undergraduate-level introduction to the theory of frames in a Hilbert space. This book can serve as a text for a special-topics course in frame theory, but it could also be used to teach a second semester of linear algebra, using frames as an application of the theoretical concepts. It can also provide a complete and helpful resource for students doing undergraduate research projects using frames.
The early chapters contain the topics from linear algebra that students need to know in order to read the rest of the book. The later chapters are devoted to advanced topics, which allow students with more experience to study more intricate types of frames. Toward that end, a Student Presentation section gives detailed proofs of fairly technical results with the intention that a student could work out these proofs independently and prepare a presentation to a class or research group. The authors have also presented some stories in the Anecdotes section about how this material has motivated and influenced their students.
Readership
Undergraduate and graduate students interested in linear algebra and applications, and the theory of frames.
Table of Contents
Introduction
Linear algebra review
Finite-dimensional operator theory
Introduction to finite frames
Frames in \mathbb{R}^2
The dilation property of frames
Dual and orthogonal frames
Frame operator decompositions
Harmonic and group frames
Sampling theory
Student presentations
Anecdotes on frame theory projects by undergraduates
Bibliography
List of symbols
Index
Student Mathematical Library, Volume: 41
2007; 175 pp; softcover
ISBN-10: 0-8218-4418-0
ISBN-13: 978-0-8218-4418-2
Expected publication date is December 23, 2007.
This book provides a brief and accessible introduction to the theory of finite fields and to some of their many fascinating and practical applications.
The first chapter is devoted to the theory of finite fields. After covering their construction and elementary properties, the authors discuss the trace and norm functions, bases for finite fields, and properties of polynomials over finite fields.
Each of the remaining chapters details applications. Chapter 2 deals with combinatorial topics such as the construction of sets of orthogonal latin squares, affine and projective planes, block designs, and Hadamard matrices. Chapters 3 and 4 provide a number of constructions and basic properties of error-correcting codes and cryptographic systems using finite fields.
Each chapter includes a set of exercises of varying levels of difficulty which help to further explain and motivate the material. Appendix A provides a brief review of the basic number theory and abstract algebra used in the text, as well as exercises related to this material. Appendix B provides hints and partial solutions for many of the exercises in each chapter. A list of 64 references to further reading and to additional topics related to the book's material is also included.
Intended for advanced undergraduate students, it is suitable both for classroom use and for individual study.
Readership
Undergraduate and graduate students interested in the theory of finite fields and applications.
Table of Contents
Finite fields
Combinatorics
Algebraic coding theory
Cryptography
Background in number theory and abstract algebra
Hints for selected exercises
References
Index
Student Mathematical Library, Volume: 42
2008; 260 pp; softcover
ISBN-10: 0-8218-4420-2
ISBN-13: 978-0-8218-4420-5
Expected publication date is January 10, 2008.
This book is an introduction to basic concepts in ergodic theory such as
recurrence, ergodicity, the ergodic theorem, mixing, and weak mixing. It
does not assume knowledge of measure theory; all the results needed from
measure theory are presented from scratch. In particular, the book includes
a detailed construction of the Lebesgue measure on the real line and an
introduction to measure spaces up to the Caratheodory extension theorem.
It also develops the Lebesgue theory of integration, including the dominated
convergence theorem and an introduction to the Lebesgue L^pspaces.
Several examples of a dynamical system are developed in detail to illustrate various dynamical concepts. These include in particular the baker's transformation, irrational rotations, the dyadic odometer, the Hajian-Kakutani transformation, the Gauss transformation, and the Chacon transformation. There is a detailed discussion of cutting and stacking transformations in ergodic theory. The book includes several exercises and some open questions to give the flavor of current research. The book also introduces some notions from topological dynamics, such as minimality, transitivity and symbolic spaces; and develops some metric topology, including the Baire category theorem.
Readership
Undergraduate and graduate students interested in ergodic theory.
Table of Contents
Introduction
Lebesgue measure
Recurrence and ergodicity
The Lebesgue integral
The ergodic theorem
Mixing notions
Set theory
The completeness property of \mathbb{R}
Topology of \mathbb{R} and metric spaces
Bibliographic notes
Bibliography
Index
Mathematical Surveys and Monographs, Volume: 144
2008; approx. 470 pp; hardcover
ISBN-10: 0-8218-4429-6
ISBN-13: 978-0-8218-4429-8
Expected publication date is January 20, 2008.
Geometric analysis has become one of the most important tools in geometry and topology. In their books on the Ricci flow, the authors reveal the depth and breadth of this flow method for understanding the structure of manifolds. With the present book, the authors focus on the analytic aspects of Ricci flow.
Some highlights of the presentation are weak and strong maximum principles for scalar heat-type equations and systems on manifolds, the classification by Bohm and Wilking of closed manifolds with 2-positive curvature operator, Bando's result that solutions to the Ricci flow are real analytic in the space variables, Shi's local derivative of curvature estimates and some variants, and differential Harnack estimates of Li-Yau-type including Hamilton's matrix estimate for the Ricci flow and Perelman's estimate for fundamental solutions of the adjoint heat equation coupled to the Ricci flow.
The authors have tried to make some advanced material accessible to graduate students and nonexperts. The book gives a rigorous introduction to some of Perelman's work and explains some technical aspects of Ricci flow useful for singularity analysis. They have also attempted to give the appropriate references so that the reader may further pursue the statements and proofs of the various results.
See also:
The Ricci Flow: An Introduction
The Ricci Flow: Techniques and Applications: Part I: Geometric Aspects
The Ricci Flow: Techniques and Applications: Part III: Geometric-Analytic Aspects (forthcoming)
Readership
Graduate students and research mathematicians interested in geometic analysis; geometry and topology.
Table of Contents
Weak maximum principles for scalars, tensors, and systems
Closed manifolds with positive curvature
Weak and strong maximum principles on noncompact manifolds
Qualitative behavior of classes of solutions
Local derivative of curvature estimates
Differential Harnack estimates of LYH-type
Perelman's differential Harnack estimate
An overview of aspects of Ricci flow
Aspects of geometric analysis related to Ricci flow
Tensor calculus on the frame bundle
Bibliography
Index