American Mathematical Society

Bifurcation theory and spatio-temporal pattern formation.

These six papers apply dynamical systems theory to problems of pattern formation in space and time, with the knowledge of the importance of nonlinear dynamical systems and the formation of spatio-temporal patterns in current research on partial differential equations. Drawn from a workshop held at the Fields Institute in December 2003 which honored William F. Langford's fundamental work in the field, the topics here include flow invariant subspaces for lattice dynamical systems, low- to high-dimensional behavior in waves in extended systems, mixed mode oscillations due to the generalized Canard phenomenon, bioremediation of waste in a porous medium, bifurcation of gyroscopic systems near a O:1 resonance, high dimensional data clustering from a dynamical systems point of view, and the computation of limit cycles as the second part of Hilbert's tenth problem. (Annotation ©2006 Book News Inc. Portland, OR)

A brief introduction to classical, statistical, and quantum mechanics.

This text contains the lecture notes for a one-semester graduate class in mechanics given by the author to first-year PhD students at the Courant Institute of Mathematical Sciences in New York. Assuming no previous knowledge of physics, the course was intended to introduce the methods of classical, statistical, and quantum mechanics to those students that may be entering the fields of applied mathematics, mathematical physics, or applied probability. The author has chosen topics that stress fundamental mechanical methods, because they provide connections between seemingly unconnected materials. (Annotation ©2006 Book News Inc. Portland, OR)

The calculus of one-sided M-ideals and multipliers in operator spaces.

The theory of one-sided M-ideals and multipliers of operator spaces is also a generalization of classical M-ideals, ideals in operator algebras, and aspects of the theory of Hilbert C*-modules and their maps. Here the authors introduce the basic concepts of one-sided multipliers and adjointable multipliers, one-sided M- and L-structure, and one-sided Cunningham algebra, continuing with spatial action such as projections, partial isometrics, Murry/Von Neumann equivalences, inner products on operator spaces, and polar decomposition. Examples include two-dimensional operator spaces, MIN and MAX spaces, Hilbertian operator spaces and locally reflexive operator spaces, and constructions include subspace and quotient, sum and intersection, minimal tensor products, interpolation, diagonal sums, and mutually orthogonal and equivalent one-sided M-projections. The authors devote chapters to one-sided type decompositions and Morita equivalence, central M-structure for operator spaces, and future directions. (Annotation ©2006 Book News Inc. Portland, OR)

Chaotic billiards.

In this new treatment of one of the most dynamic but difficult topics in modern theory, Chernov and Markarian keep the beginner in mind as they start from the basics and work through all the definitions and give full proofs of the main theorems as they cover basic constructions, Lyapunov exponents and hyperbolicity, dispersing billiards, dynamics of unstable manifolds, ergodic properties, statistical properties, Bunimovich billiards and general focusing chaotic billiards. Readers should have completed graduate courses in measure theory, probability, Riemann geometry, topology and complex analysis. (Annotation ©2006 Book News Inc. Portland, OR)

Complex graphs and networks.

Graph theory has become a primary tool for detecting hidden structures in information networks, including Internet graphs, social and biological networks, or any situation requiring large data sets. Based on ten lectures given at the CBMS Workshop on the Combinatorics of Large Sparse Graphs at California State U. at San Marcos in June 2004, this covers the basics of graph theory, including degree sequences and the power law, old and new concentration inequalities, including the work of Chernoff, Martingale and Azuma, the preferential attachment scheme as a generative model, duplication models for biological networks, random graphs and given expected degrees, the rise of the "giant component," average distance and the diameter, Eigenvalues of the adjacent matrix of G(w) and the semi-circle law, coupling online and off-line analyses of random graphs, the configuration model for power law graphs, and the "small world" phenomenon in hybrid graphs. (Annotation ©2006 Book News Inc. Portland, OR)

Differential geometry; curves - surfaces - manifolds, 2d ed.

Writing for undergraduates with courses in analysis and linear algebra, Kuhnel works carefully and logically through his chapters, giving newcomers a summary of the prerequisite analysis and moving directly to curves, including plane curves and space curves, Frenet curves, and curves in Minkowski space, the local theory of surfaces, including surface elements, first fundamental form, the Gauss map and the curvature of surfaces, surfaces of rotation and rules surfaces, minimal surfaces and surfaces in Minkowski space, the intrinsic geometry of surfaces, including the covariant derivative, the Gaussian equation and the Theorema Egregium, and the Gauss-Bonnet theorem, Riemann manifolds, including the tangent space and the Reimannian connection, the curvature tensor, including the Ricci and Einstein tensors, spaces of constant curvature, including geodesics and Jacobi fields, and Einstein spaces, including the decomposition of the curvature tensor and the Weyl tensor. The text is clearly and carefully illustrated. (Annotation ©2006 Book News Inc. Portland, OR)

Disease evolution; models, concepts, and data analyses.

This work introduces scientists and researchers to current techniques and challenges in mathematical evolutionary epidemiology. It expands on material presented at the November 2003 DIMACS working group on Genetics and Evolution of Pathogens, held at Rutgers University, and includes papers from experts who were unable to attend the meeting. Contributors approaching the study of epidemiology from a variety of applied and theoretical disciplines, including mathematics, computer science, statistics, and biology, outline general concepts underlying models of disease evolution, and look at methodological challenges posed by factors such as drug resistance and host dynamics. They show how methods are used to investigate the evolution of specific diseases, including HIV and malaria. There is no subject index. (Annotation ©2006 Book News Inc. Portland, OR)

Floer homology, gauge theory, and low dimensional topology; proceedings.

The actual lectures at the course were tailored to fit into a curriculum that included problem sessions and tutoring, and so have been revised somewhat for the published volume. The 13 papers cover Heegaard Floer homology and knot theory, Floer homologies and contact structures, and symplectic four-manifolds and Seiberg-Witten invariants. They are not indexed. (Annotation ©2006 Book News Inc. Portland, OR)

Galois theory for beginners; a historical perspective.

Writing for advanced undergraduates and early graduate students, Bewersdorff includes the material necessary to understand groups and fields within his text. With concrete examples he begins with cubic equations, turning to complex numbers, biquadratic equations, equations of degree n and their properties, including plausibility and proof, the search for additional solution formulas, equations that can be reduced in degree, including the decomposition of integer polynomials and Eisenstein's irreducibilty criterion, the construction of regular polygons, the Galois group of an equation, and algebraic structures and Galois theory, including groups and fields, the fundamental theorem, and Artin's version of the fundamental theory. (Annotation ©2006 Book News Inc. Portland, OR)

Graduate algebra; commutative view.

Rowen considers students at various levels of mathematical understanding and works to bring harmony to the interplay between algebra and geometry and the study of algebraic structures through their representations into matrices, calling into play linear algebra. He introduces modules, then covers finely integrated modules and their application to Abelian groups, simple modules and composition series. In his treatment of affine algebras and Noetherian rings he describes the Galois theory of fields, algebras and affine fields, transcendence degree and the Krull dimension of a ring, modules and rings satisfying chain conditions, localization in the prime spectrum, the Krull dimension theory of commutative Noetherian rings. In his section on applications to geometry and number theory he covers the algebraic foundations of geometry, applications to algebraic geometry over the rations, including diophantine equations and elliptic curves, and absolute values and valuation rings. He provides exercises and a list of major results. (Annotation ©2006 Book News Inc. Portland, OR)

Harmonic analysis; Calderón-Zygmund and beyond; proceedings.

Eight papers consider not only harmonic analysis itself, but also its application to other areas of mathematics, particularly partial differential equations and ergodic theory. Among the topics are fractional calculus associated with doubling and non-doubling measures, some recent developments in the well-posedness of non- linear dispersive equations, and variation inequalities for singular integrals and related operators. Several questions raised in the Problems session are also presented. (Annotation ©2006 Book News Inc. Portland, OR)

Homotopy theory of schemes.

Looking at the category of schemes from a homotopic perspective, Morel, who is not further identified, defines, for all reasonable schemes, the homotopy category of smooth category-schemes, and argues that this category plays the same role for smooth category-schemes as it does for the classical homotopic category for differentiable varieties. Théorie homotopique des schémas was published in 1999 by Sociéte Mathématique de France. It is not indexed. (Annotation ©2006 Book News Inc. Portland, OR)

Inverse problems, multi-scale analysis, and effective medium theory; proceedings.

Papers from a June 2005 conference describe new techniques for solving inverse problems, with emphasis on their connection to multi- scale analysis and the mathematical theory of composite materials. Topics are addressed from analytic, numerical, and physics perspectives. The methods used to solve inverse problems come from a range of areas of pure and applied mathematics, including potential theory, partial differential equations, scattering theory, complex analysis, and numerical methods. The book is of interest to mathematicians, physicists, and engineers, and to researchers and graduate students working in partial differential equations and applications. (Annotation ©2006 Book News Inc. Portland, OR)

Ischia group theory 2004; proceedings.

This proceedings volume contains papers presented at the Israel Mathematical Conference, which was held in Naples, Italy in 2004 in honor of Marcel Herzog (retired, Tel-Aviv U.). The 17 articles deal with various aspects of group theory — Herzog's primary area of research. A sampling of topics includes monounary simple algebras; group rings with simple augmentation ideals; and the theory of finite 2-groups. The volume is not indexed. (Annotation ©2006 Book News Inc. Portland, OR)

Lectures and exercise on functional analysis.

Functional analysis is the branch of mathematics concerned with the study of spaces of functions. Helemskii (Moscow State U., Russia) presents this text as an introduction to the field, devoting the bulk of it to the classical functional analysis of normed, Banach, and Hilbert spaces, but also covering polynormed spaces and some of their applications, as well as those elements of the theory of Banach algebras important to the study of spectra. The treatment differs from others on the same topic in that it emphasizes the categorical nature of the fundamental construction and results. The text assumes knowledge of linear algebra, the foundations of real analysis, and the elements of the theory of metric spaces found in advanced analysis courses. (Annotation ©2006 Book News Inc. Portland, OR)

Lecture notes on motivic cohomology.

Mazza and Charles Weibel wrote the book to reflect a series of lectures that Vladimir Voevodsky delivered to graduate mathematics students at the Institute for Advance Study in Princeton, New Jersey, in 1999-2000. The 24 lectures cover presheaves with transfers, Étale motivic theory, Nisnevich sheaves with transfers, the triangulated category of motives, higher Chow groups, and Zariski sheaves with transfers. (Annotation ©2006 Book News Inc. Portland, OR)

Lie groups and antomorphic forms; proceedings.

In these proceedings of the Summer 2003 international annual meeting, contributors describe their work in such fields as differential geometry, algebraic geometry, representation theory, number theory, and other areas. The lectures' topics include Lie groups and linear algebraic groups in terms of complex and real groups, the cohomology of arithmetic groups, local symmetric spaces and arithmetric groups, Petersson and Kuznetsov trace formulas, and the cohomology of locally symmetric spaces and of their compactifications. (Annotation ©2006 Book News Inc. Portland, OR)

Mathematical ciphers; from Caesar to RSA.

Beginning with the encryption system used by Julius Caesar, Young (Loyola College) explains ever more complicated schemes for coding messages, culminating in the RSA Cipher developed by MIT computer scientists for internet security. The undergraduate textbook introduces number theory, modular arithmetic, substitution ciphers, the Euclidean algorithm, and the mathematical basis for an exponential cipher. (Annotation ©2006 Book News Inc. Portland, OR)

Measure theory and integration.

Taylor (U. of North Carolina) presents material he has often used in a one-semester course on measure theory for students who have completed an introductory analysis course covering such matters as metric spaces, the uniform convergence of functions, the contraction mapping principles, and aspects of multi-variable calculus. He includes chapter-end exercises. (Annotation ©2006 Book News Inc. Portland, OR)

Number theory in the spirit of Ramanujan.

Suitable for advanced undergraduates or beginning graduate students, this introduction to Ramanujan's work in q-series and theta functions also sets this greatest of number theorists within the context of his world in and out of mathematics. Berndt keeps in the spirit of Ramanujan in his proofs (Ramanujan did not record his own) and gives a summary of the results established for each chapter, placing them in historical and contemporary contexts. He covers Ramanujan's work in congruences for p(n) and r(n), sums of squares and sums of triangular numbers, Eisenstein series, the connection between hypergeometric functions and theta functions and the applications of the primary theorem, and the Rogers-Ramanujan continued fraction. The resulting text is concise, rigorous, elegant and exhilarating in its explanations of Ramanujan's invisible proofs. (Annotation ©2006 Book News Inc. Portland, OR)