IAS/Park City Mathematics Series, Volume: 16
2009; 360 pp; hardcover
ISBN-13: 978-0-8218-4671-1
Expected publication date is July 3, 2009.
In recent years, statistical mechanics has been increasingly recognized as a central domain of mathematics. Major developments include the Schramm-Loewner evolution, which describes two-dimensional phase transitions, random matrix theory, renormalization group theory and the fluctuations of random surfaces described by dimers. The lectures contained in this volume present an introduction to recent mathematical progress in these fields. They are designed for graduate students in mathematics with a strong background in analysis and probability.
This book will be of particular interest to graduate students and researchers interested in modern aspects of probability, conformal field theory, percolation, random matrices and stochastic differential equations.
Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.
Graduate students and research mathematicians interested in probability and its applications in mathematics and physics.
D. C. Brydges -- Lectures on the renormalisation group
A. Guionnet -- Statistical mechanics and random matrices
R. Kenyon -- Lectures on dimers
G. Lawler -- Schramm-Loewner evolution $(SLE)$
W. Werner -- Lectures on two-dimensional critical percolation
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Student Mathematical Library, Volume: 48
2009; approx. 205 pp; softcover
ISBN-13: 978-0-8218-4862-3
Expected publication date is July 8, 2009.
This book provides a student's first encounter with the concepts of measure theory and functional analysis. Its structure and content reflect the belief that difficult concepts should be introduced in their simplest and most concrete forms.
Despite the use of the word "terse" in the title, this text might also have been called A (Gentle) Introduction to Lebesgue Integration. It is terse in the sense that it treats only a subset of those concepts typically found in a substantial graduate-level analysis course. The book emphasizes the motivation of these concepts and attempts to treat them simply and concretely. In particular, little mention is made of general measures other than Lebesgue until the final chapter and attention is limited to $R$ as opposed to $R^n$.
After establishing the primary ideas and results, the text moves on to some applications. Chapter 6 discusses classical real and complex Fourier series for $L^2$ functions on the interval and shows that the Fourier series of an $L^2$ function converges in $L^2$ to that function. Chapter 7 introduces some concepts from measurable dynamics. The Birkhoff ergodic theorem is stated without proof and results on Fourier series from Chapter 6 are used to prove that an irrational rotation of the circle is ergodic and that the squaring map on the complex numbers of modulus 1 is ergodic.
This book is suitable for an advanced undergraduate course or for the start of a graduate course. The text presupposes that the student has had a standard undergraduate course in real analysis.
Undergraduate students interested in topology and/or geometry of low-dimensional manifolds, particularly 3-manifolds.
The regulated and Riemann integrals
Lebesgue measure
The Lebesgue integral
The integral of unbounded functions
The Hilbert space $L^2$
Classical Fourier series
Two ergodic transformations
Background and foundations
Lebesgue measure
A non-measurable set
Bibliography
Index
Graduate Studies in Mathematics, Volume: 104
2009; approx. 728 pp; hardcover
ISBN-13: 978-0-8218-4781-7
Expected publication date is July 19, 2009.
Algebra: Chapter 0 is a self-contained introduction to the main topics of algebra, suitable for a first sequence on the subject at the beginning graduate or upper undergraduate level. The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics. A second feature consists of an emphasis on homological algebra: basic notions on complexes are presented as soon as modules have been introduced, and an extensive last chapter on homological algebra can form the basis for a follow-up introductory course on the subject. Approximately 1,000 exercises both provide adequate practice to consolidate the understanding of the main body of the text and offer the opportunity to explore many other topics, including applications to number theory and algebraic geometry. This will allow instructors to adapt the textbook to their specific choice of topics and provide the independent reader with a richer exposure to algebra. Many exercises include substantial hints, and navigation of the topics is facilitated by an extensive index and by hundreds of cross-references.
Undergraduate and graduate students interested in algebra.
Preliminaries: Set theory and categories
Groups, first encounter
Rings and modules
Groups, second encounter
Irreducibility and factorization in integral domains
Linear algebra
Fields
Linear algebra, reprise
Homological algebra
Index
Series: Trends in Mathematics
2009, Approx. 350 p., Hardcover
ISBN: 978-3-7643-9910-8
This volume assembles several research papers in all areas of geometric and combinatorial group theory originated in the recent conferences in Dortmund and Ottawa in 2007. It contains high quality refereed articles developping new aspects of these modern and active fields in mathematics. It is also appropriate to advanced students interested in recent results at a research level.
Graduate students and researchers in geometric and combinatorial group theory and related areas
Dynamics of free group automorphisms.- Statistical questions in group theory motivated by cryptography.- Solving random equations in Garside groups using length functions.- Limits of (Thompsonfs) group F.- Twisted conjugacy for virtually cyclic groups and crystallographic groups.- Generating tuples of virtually free groups.- Unsolvability of the isomorphism problem for [free-abelian]-by-free groups.- An application of word combinatorics to detection problems in group theory.- The Fn-action on the product of the two limit trees for an iwip automorphism.- A context-sensitive combing associated with Baumslag-Solitar 2,7.- Algebraic geometry over additive positive monoids. Systems of coefficient-free equations.- Some graphs related to Thomsonfs group F.- Subgroups of small index in the automorphism groups of free groups and the (T) property of Kazhdan
Series: Applied and Numerical Harmonic Analysis
2009, Approx. 310 p. 50 illus., Hardcover
ISBN: 978-0-8176-4879-4
This textbook presents the mathematics that is foundational to multimedia applications. Featuring a rigorous survey of selected results from algebra and analysis, the work examines tools used to create application software for multimedia signal processing and communication.
* Over 100 exercises with complete solutions
* Many sample programs in Standard C
* Numerous illustrations based on data from real studies
* Suggestions for further reading at the end of each chapter
* A companion website providing the computer programs described in the book as well as additional references and data files, such as images and sounds, to enhance the readerfs understanding of key topics
* Only undergraduate-level knowledge of mathematics and no previous knowledge of statistics are required
Mathematics for Multimedia is an ideal textbook for upper undergraduate and beginning graduate students in computer science and mathematics who seek an innovative approach to rigorous, contemporary mathematics with practical applications. The work may also serve as an invaluable reference for multimedia applications developers and all those interested the mathematics underlying multimedia design and implementation.
Upper undergraduate and beginning graduate students in computer science and mathematics; multimedia applications developers and all those interested in the mathematics underlying multimedia design and implementation
Numbers and Arithmetic.- Space and Linearity.- Time and Frequency.- Sampling and Estimation.- Scale and Resolution.- Redundancy and Information.- Answers.- Basics, Technicalities, and Digressions.- Index.
Series: Progress in Mathematical Physics , Vol. 56
2009, Approx. 350 p., Hardcover
ISBN: 978-3-7643-9970-2
Physically intuitive, but mathematically rigorous writing style
Early introduction to geometric quantities allows students to learn simpler topics in general relativity (Newtonian limit, Schwarzschild solution, bending of light in a gravitational field, ...)
Approved in lectures given by the author for several years
Teaching Einsteinfs general relativity at introductory level poses problem because students cannot begin to appreciate the basics of the theory unless they learn a sufficient amount of Riemannian geometry. Most elementary books take the easy course of telling the students a few working rules stripping the mathematical details to a minimum while the advanced books take the mathematical background for granted. Students eager to study Einsteinfs theory at a deeper level are forced to learn the mathematical background on their own and they feel lost because pure mathematical texts on geometry are too abstract and formal.
The present book solves the pedagogical problem in a unique way by dividing the book in three parts. Essential concepts of Riemannian geometry are introduced in Part I (four chapters) through Gaussf work on curvature of surfaces using only ordinary calculus. A first acquaintance with Einsteinfs theory can then be made. Only after this first brush with both physics and mathematics of relativity, a proper, detailed mathematical background is developed in the next six chapters in Part II. The third part then recaptures all the basic concepts of general relativity and leaves the student with a sound preparation for learning advanced topics.
My aim has been that after learning from this book a student should not feel discouraged when she opens advanced texts on general relativity for further reading.
I.Spacetime: 1. Introduction.- 2. What is Curvature?.- 3. General relativity basics.- 4. Spherically symmetric gravitational field.- II. Geometry: 5. Vectors and Tensors.- 6. Inner Product.- 7. Elementary Differential Geometry.- 8. Connection and Curvature.- 9. Riemannian geometry.- 10. Some More Geometry.- III. Gravitation: 11. The Einstein Equation.- 12. General features of spacetime.- 13. Weak Gravitational fields.- 14. Schwarzschild and Kerr solutions.- 15. Cosmology.