American Mathematical Society

Additive combinatorics; proceedings.

This work collects written versions of most of the lectures given at a spring 2006 school held at the University of Montreal, Canada, plus submitted contributions from speakers at a related workshop. Some specific papers topics addressed include bounds on exponential sums over small multiplicative subgroups, ergodic methods in additive combinatorics, the ergodic and combinatorial approaches to Szemerédi's theorem, problems related to sum-product theorems, additive properties of product sets in fields of prime order, and open problems in additive combinatorics. The book will be of interest to mathematicians from many different backgrounds, including group theory, complexity theory, discrete geometry, and ergodic theory. There is no subject index. (Annotation ©2007 Book News Inc. Portland, OR)

Algebraic geometry codes; basic notions.

This graduate textbook introduces the theory of error-correcting codes, working over an arbitrary finite field; algebraic curves, working over an algebraically closed field; geometry over a finite field; and constructions of algebraic geometry codes. The Russian authors discuss the Riemann-Roch theorem, the theory of elliptic curves, asymptotic bounds for the number of points on a curve, calculation of the number of divisors with prescribed properties, towers of curves with many points, and asymptotic lower bounds of algebraic geometry origin. (Annotation ©2007 Book News Inc. Portland, OR)

Analytic number theory; a tribute to Gauss and Dirichlet; proceedings.

Work presented here is based on talks given at a June 2005 conference commemorating the 150th anniversary of the death of C.-F. Gauss and the birth of J.-L. Dirichlet. The book begins with a summary of the life and work of Dirichlet, and continues with 13 papers on current analytic number theory research topics that were directly influenced by Gauss and Dirichlet. Coverage encompasses recent advances in both classical analytic number theory, as well as advances in related parts of number theory and algebraic geometry. Among the topics examined are the distribution of primes (long arithmetic progressions of primes and small gaps between primes), class groups of binary quadratic forms, various aspects of the theory of L-functions, the theory of modular forms, and the study of rational and integral solutions to polynomial equations in several variables. There is no subject index. (Annotation ©2007 Book News Inc. Portland, OR)

Approximate approximations.

Writing for graduate students and researchers with experience in functional analysis and numerical methods, the authors describe applications for new procedures of approximation, most of which include approximate quasi-interpolation, interpolation, least square approximation, cubature of integral operations and wavelet approximations. These procedures have the one common feature of being accurate without being convergent in a rigorous sense. They begin with quasi-interpolation on uniform lattices and describe methods of estimating error in quasi-interpolation, then describe a variety of basis functions and algorithms for their construction, semi-analytic cubature formulas for integral and pseudodifferential operators of mathematical physics (including harmonic, elastic and diffraction potentials), approximations of the inverse operator of the Cauchy problem (for heat, wave and plate equations), quasi-interpolation and interpolation of Gaussian functions, approximate wavelets, cubature over bounded domains, general grids, scattered data approximate approximations, and non-linear and linear problems with numerical algorithms based upon approximate approximations. (Annotation ©2007 Book News Inc. Portland, OR)

Decompositions of manifolds.

Decompositions, also called partitions create cell-like sets. They are compact sets, similar to those that are contractable, and behave homotropically much like points. Daverman builds on the work of Moore in the 1920s and Bing in the 1950s as well as Edwards and Quinn as he progresses to higher-dimensional anifolds in terms of elementary topological properties. Daverman covers the preliminaries in terms of elementary properties of upper semicontinuous decompositions, upper semicontinuous decompositions, proper maps and monotome decompositions, the shrinkabilty criterion, cell-like decompositions of absolute neighborhood retracts, the cell-like approximation theorem, shrinkable decompositions, nonshrinkable decompositions, and applications to manifolds. The 41 chapters work well in the classroom but also would work in self-study. (Annotation ©2007 Book News Inc. Portland, OR)

Episodes in the history of modern algebra (1800-1950).

No oil has been discovered in algebra, so it has not been in the headlines for some time, but still changes have been going on. Mathematicians and historians of mathematics recount some of those changes over a century and a half. Among them are Duncan F. Gregory, the calculus of operations, and the Cambridge Mathematical Journal; developments in the theory of algebras over number fields; Emmy Noether's 1932 ICM lecture on non-commutative methods in algebraic number theory; and the arithmetization of algebraic geometry. The 13 papers are from an April 2003 workshop in Berkeley. (Annotation ©2007 Book News Inc. Portland, OR)

Filtering and prediction; a primer.

The authors stick to the basics in describing the process of observing moving objects when the observations are corrupted by random errors, filtering out the errors and extracting the most precise information possible, and then predicting the future path of the object from past behavior. They do so as accessibly as possible without significantly reducing rigor, giving students a solid background in such prerequisites for studying filtering and prediction as series, probability concepts and conditioning in the discrete case. With plenty of examples and exercises, they cover Markov chains, filtering of discrete Markov chains, conditional expectations, filtering of continuous-space Markov chains, Wiener process and continuous time filtering, stationary sequences and prediction of stationary sequences. The result is a handy guide not only for students but for nonspecialist professionals or those who need to review basic concepts. (Annotation ©2007 Book News Inc. Portland, OR)

Fluids and waves; recent trends in applied analysis; proceedings.

This lively field has come to include the study of partial differential equations, dynamical systems and stochastic processes as well as classical analysis, and has applications ranging from engineering to neuroscience, physics and biology. These 17 articles, based on lectures from the May 2006 conference, include both techniques in analysis and applications and offer links between current research themes. Specific topics include the vibrations of a damageable string, an explicit semigroup generator and its spectral properties, energy-level weak solutions for a nonlinear fluid-structure interaction model, stimulus-induced bumps in two-dimensional neural field theory, exotic dynamics in a firing rate model of neural tissue with threshold accommodation, embedded models of self-assembly in DNA complexes, stochastic models with negative friction for intermittent rolling of biological mimetics, neural networks that learn to approximate autonomous flows, the method of Lyapunov functions of two variables, and an analysis of map formation in visual perception. (Annotation ©2007 Book News Inc. Portland, OR)

Function spaces; proceedings.

Drawn from those presented at the May 2006 conference, some of these 33 papers contain open problems, others contain an exposition of results known to experts, and others contain new discoveries that should be accessible to mathematicians working in different areas of function spaces. Specific topics include Young-Fenchel transformation and some new characteristics of Banach spaces, an example of the boundary of topologically inverted elements, sums and products of bad functions, disc algebra and a moment problem, the stability of logmodularity for uniform algebras, regularity and amenability conditions for uniform algebras, closed suns of marginal subspaces of Banach function space, surjections on the algebras of continuous functions which preserve peripheral spectrum, asymptotics of Toeplitz determinants generated by functions with Fourier coefficients in weighted Orlicz sequence classes, spectral isometries, examples of Banach spaces that are not Banach algebras, and a spectra of algebras of analytic functions and polynomials on Banach space. (Annotation ©2007 Book News Inc. Portland, OR)

Harmonic analysis on commutative spaces.

Drawing from his lectures at the U. of California, Berkeley conducted over a period of years, Wolfe focuses on the analytic approach rather than the geometric approach to the theory of commutative spaces, noting that recent work in the latter has reached a certain plateau, creating an appropriate moment for this monograph. He describes commutative space theory as a common generalization of the theories of compact topological groups, compact Abelian groups, Riemannian symmetric spaces and multiply transitive transformation groups, and divides his examination into basic information for first-year graduate students (on general theories of topological groups, representation theory and compact groups), intermediate information on commutative spaces for those with a bit of analytic sophistication and a section on structure and analysis for commutative spaces for mathematicians entering their research careers who want to see how far they can go with this unified view of algebra, geometry and analysis. (Annotation ©2007 Book News Inc. Portland, OR)

Homogenization; methods and applications.

Chechkin (Lomonosov Moscow State University) details partial differential equation techniques that are used to study differential operators with rapidly oscillating coefficients, equations in perforated domains, and equations with random coefficients. Focus is on various aspects of homogenization theory and related topics. Coverage encompasses classical results and methods of homogenization theory, as well as techniques developed in the last decade. Special attention is paid to averaging of random parabolic equations with lower order terms, and to boundary value problems with rapidly varying boundary conditions. Simple and advanced examples and exercises are included. The book was originally published in Russian in 2007, and is translated by Tamara Rozhkovskaya. (Annotation ©2007 Book News Inc. Portland, OR)

Hopf algebras and generalizations; proceedings.

Work in this interdisciplinary field has revealed structures connected to various areas of mathematics, particularly through quantum groups. It locates methods, relationships, motivations and applications throughout algebra, category theory, topology, geometry, quantum field theory, quantum gravity, combinatories, logic and theoretical computer science. These proceedings for the October 2004 special session include eight papers that explore essential aspects of Hopf algebras and some of their best-known generalizations through a variety of approaches and perspectives; topics include lax braiding and the lax center, dynamical quantum groups, groups of group-like elements of a semi-simple Hopf algebra and its dual, higher Fobenius-Schur indicators for pivotal structures, doubles of (quasi) Hopf algebras and some examples of quantum groupoids and vertex groups related to them, central braided Hopf algebras, anti-Yetter-Drinfield algebras and representations of the Hopf algebra U(n). (Annotation ©2007 Book News Inc. Portland, OR)

Interactions between homotopy and algebra; proceedings.

Based on the lectures presented at the July/August 2004 summer school at the U. of Chicago, and aimed at beginning researchers with varied mathematical backgrounds, these articles create a background in the discipline and descriptions of current directions of research related to deep connections between homotopy theory and algebra, including algebraic geometry, commutative algebra and representation theory. Topics of introductory lectures include model categories and simplical methods, local cohomology, derived categories, resolutions, and Brown representability. Topics of the lecture series include spectra for commutative algebra, rational homotopy theory, André-Quillen homology of commutative algebras, first steps in commutative algebra, cotorsion pairs and model categories, coherent sheaves on an elliptical curve, and the cohomology of finite groups. (Annotation ©2007 Book News Inc. Portland, OR)

Introduction to complex analysis, 2d ed.

This text is based on lectures given by the authors at Helsinki University and at the University of Zürich, and is a translation of the German edition, Enidühung in die Funktionentheorie, published by Birkhäuser Verlag, Basel, in 1964. The book is limited to elements of the theory of functions. Coverage includes complex numbers, the concept of homotopy and its application, integral theorems, the theory and application of harmonic functions, and the correspondence of boundaries under conformal mapping. In introducing elementary functions, the authors have followed the presentation given by Ernst Lindelöf in his Finnish textbook, Johdatus funktioteoriaan. Chapter exercises are included, for a total of 320 exercises. It is assumed that the reader is acquainted with analytic geometry and calculus. (Annotation ©2007 Book News Inc. Portland, OR)

Lie algebras, vertex operator algebras, and their applications; an international conference in honor of James Lepowsky and Robert L. Wilson on their sixtieth birthdays, May 17-21, 2005, North Carolina State University, Raleigh, North Carolina.

Together or separately, Lepowsky and Wilson have made enormous contributions to the theory of Lie algebras and the theory of vertex operator algebras. It is right their students and colleagues honor them simultaneously with this collection of articles based on talks conducted at the conference. Most focus on Lie algebras (of course) but others also cover quantum groups, vertex operator algebras and their applications in number theory, combinatorics and conformal field theory. Topics of the more than two dozen papers include a class of gradings of simple Lie algebras, support spaces and Auslander-Reiten components, the cohomology of modular Lie algebras, Maas-Selberg relations, new constructions of Yang-Baker systems, Lie algebras of small dimensions, intertwining operators and recursions, representations of vertex operator algebras, twisted modules and quasi-modules, chiral algebras and partition functions and Hilbert schemes on points on the minimal resolution and Soliton equation. (Annotation ©2007 Book News Inc. Portland, OR)

Mathematical omnibus; thirty lectures on classic mathematics.

The 30 lectures are derived from articles that Fuchs and Tabachnikov wrote for the Russian magazine Kvant 1970-90 and from lectures they gave in the Soviet Union and the US to audiences that included advanced high school and undergraduate mathematics students. The collection can be used for formal or informal teaching or slow self-study at those levels. It is studded with explanatory illustrations by Sergey Ivanov and with photographs of mathematicians. (Annotation ©2007 Book News Inc. Portland, OR)

Modeling and simulation of biological networks; proceedings.

The first of the five presentations delivered during the January 2006 course discuss mathematical tools that help analyze genome-level information, locate genes in newly sequenced genomes, and organize evolutionary information. The remaining chapters describe the biochemical networks that translate genome-level information into cellular metabolism, a control-theoretic approach to problems in immunology, and an interactive-based approach to modeling population biology and epidemiology. (Annotation ©2007 Book News Inc. Portland, OR)

Prospects in mathematical physics; proceedings.

This work contains 13 papers presented at the Young Researchers Symposium of the July 2003 14th International Congress on Mathematical Physics, including papers by two of the three winners of the meeting's Henri Poincaré Prize. The goal of the book is to illustrate various promising areas of mathematical physics in a way that is accessible to researchers at the beginning of their careers. Paper topics include equilibrium statistical mechanics of quantum lattice systems, the rate of local equilibration in kinetic theory, bifurcations of relative equilibria in simple mechanical systems, and generating spectral gaps by geometry. Other subjects are string theory and knot invariants, quantum geometry of Riemann surfaces, and scaling algebras for charge carrying quantum fields and superselection structure at short distances. There is no subject index. (Annotation ©2007 Book News Inc. Portland, OR)

Recent advances in nonlinear partial differential equations and applications; proceedings.

Records 11 of the many talks delivered during the June 2006 conference honoring two New York University mathematicians. The speakers examine Brownian motion in a wedge, the semiclassical focusing nonlinear Schrodinger equation, the dynamics of turbulent flows near smooth walls, the incompressible Navier-Stokes limit of the Boltzmann equation, and hyperbolic conservation laws with involutions and contingent entropies. The title is printed on the spine without the word "recent". No index is provided. (Annotation ©2007 Book News Inc. Portland, OR)

Recent developments in nonlinear partial differential equations; proceedings.

This work contains research and expository articles based on talks presented at the 2nd Symposium on Analysis and PDEs, held at Purdue University. The Symposium focused on topics related to the theory and applications of nonlinear partial differential equations which are at the forefront of current international research. Subjects featured include kinetic formulations of nonlinear PDEs, constrained Hamilton-Jacobi equations, nonlinear Schrödinger equations, quasiminimal sets for Hausdorff measures, Schrödinger flows into Kähler manifolds, and parabolic obstacle problems with applications to finance. The concise presentation makes the book suitable for both researchers and graduate students. There is no subject index. (Annotation ©2007 Book News Inc. Portland, OR)

Ricci flow and the Poincarré conjecture.

Inspired by Grigory Perelman's three preprints, although not slavishly devoted to Perelman's organization, Morgan and Tian provide full details of a complete proof of geometric arguments (particularly the Ricci flow as introduced and studied by Hamilton) that establish the Poincaré Conjecture in the affirmative. In this comprehensive and well-illustrated monograph they describe the background of Riemann geometry and Ricci flow, including manifolds of non-negative curvature, the maximum principle and convergence results for Ricci flow, Perelman's length function and its applications, including a comparative geometry approach to the Ricci flow, complete Ricci flows of bounded curvature, results and the standard solution, Ricci flow with surgery, and completion of the proof of the conjecture, including finite-time extinction. An appendix describes 3-manifolds covered by canonical neighborhoods. (Annotation ©2007 Book News Inc. Portland, OR)

Topology and robotics; proceedings.

Twelve papers from the July 2006 workshop explore the topology of certain classes of configuration spaces, the topological complexity of motion planning, and minimal sensing algorithms for robot navigation. Topics include winding numbers for networks with weak angular data, diffusion over tensor fields via Lie group PDE flows, homotopy meaningful hybrid model structures, cohomology rings of tree braid groups, geometric descriptions of polygon and chain spaces, and symmetric motion planning. No index is provided. (Annotation ©2007 Book News Inc. Portland, OR)

Topics in harmonic analysis and ergodic theory; proceedings.

Based on talks delivered at a December 2005 conference, this monograph considers the interplay between ergodic theory and harmonic analysis, and reviews the work of Roger Jones and Marshall Ash at DePaul University. Topics of the ten chapters include the central limit theorem for random walks, ergodic theory structures for foliations, uniqueness questions for multiple trigonometric series, the s-function and exponential integrability, and Nehari's theorem on the polydisk. No index is provided. (Annotation ©2007 Book News Inc. Portland, OR)