American Mathematical Society

Adventures in mathematical physics; proceedings.

The results of a very lively September 2006 conference, these 18 papers describe active research in quantum and statistical mechanics including topics ranging from open quantum systems and random Schrödinger operators to random matrix theory. Dedicating their papers to Combes and his work on transport and spectral problems in quantum mechanics, contributors address topics such as the emptiness formation probability in quasi-free states, mutually unbiased bases, perturbative and variational methods in the study of carbon nanotubes, normal transport at positive temperatures in classical Hamiltonian open systems, optimal uniform elliptic estimates for the Ginzburg-Landau system, generalized fractal dimensions on the negative axis for non-compactly supported measures, vibrational levels associated with hydrogen bonds and semi-classical Hamiltonian normal forms, the strict positivity of entropy production, heat kernels on metric graphs and a trace formula, revivals of wave packets and Bohr-Sommerfeld quantization rules, and exponential decay of eigenfunctions of first order systems. (Annotation ©2008 Book News Inc. Portland, OR)

Algebra, geometry and their interactions; proceedings.

These proceedings from the October 2005 conference include 18 papers that reflect the cutting edge of present research in algebraic geometry, commutative algebra, and numerical analysis with a focus on the benefits of interaction amongst these fields. Paper topics include the projective reconstruction from on-view near critical configurations in higher dimensions, examples and counterexamples of the conjectured Hilbert functions of multiple points, ideals in Noetherian rings, cohomological degrees and the HomAB conjecture, generating sets for the first syzygies of a monomial ideal, resolutions of square-free monomial ideals via facet ideals, properties of Lyubeznik's F-modules, ample vector bundles with sections vanishing on submanifolds of sectional genus three, finding all real points of a complex curve, the multiplicity conjecture, detecting torsion points and subtori, divisor class groups of grades hypersurfaces, irrelevant exceptional divisors for curves on a smooth surface, variations of hyperplane sections and jet schemes of determinantal varieties. (Annotation ©2008 Book News Inc. Portland, OR)

Algebra with Galois theory.

Artin's approach begins with his analysis of groups, including the concept of a group itself and the phenomena of subgroups. He describes rings and fields, including linear equations in a field and vector spaces, polynomials over a field, factorization into primes, ideals and the greatest common divisor, solution of the general equation of nth degree, residual classes, extension fields, and isomorphisms. He explains Galois theory, including such topics as splitting fields and their automorphisms, the characteristics of a field, derivation of a polynomial (multiple roots), the degree of an extension field, group characters, fundamental theorems and finite fields, then moves to polynomials with integral coefficients, including irreducibility and primitive roots of unity, and the theory of equations, including ruler and compass constructions, and the theorems of Steinitz and Abel. This classic 1947 treatment, formerly titled Modern Higher Algebra Galois Theory, has been annotated by Albert A. Blank. (Annotation ©2008 Book News Inc. Portland, OR)

Applied linear algebra; the decoupling principle, 2d ed.

Writing for junior and senior undergraduates, Sadun (mathematics, U. of Texas at Austin) offers diagonalization (reducing coupled linear evolution problems to a sequence of simpler decoupled problems) as a significant tool suitable for physics and engineering as well as in mathematics. He explains vector spaces and bases, linear transformations and operators, eigenvalues, circles and ellipses, inner products, adjoints, Hermitian operators, unitary operators, the wave equation, continuous spectre and the Dirac delta function, Fourier transforms, Green's and functions, and includes an appendix on matrix operations (new to this edition) and a full chapter on crucial applications. For this edition he also includes new exercises (with solutions), and a series of very good numerical labs, some of which are theoretical and relate linear algebra to quantum physics. Sadun also thoughtfully supplies a section for instructors, including recommendations for course outlines. (Annotation ©2008 Book News Inc. Portland, OR)

Asymptotic analysis and singularities; hyperbolic and dispersive PDEs and fluid mechanics.

Volume one of the two-volume set contains 24 papers presented at the July 2005 meeting held in Japan on nonlinear partial differential equations, dynamical systems, calculus of variations, and mathematical physics. Topics include uniform decay estimates for the wave equation in an exterior domain, Helmholtz decomposition in Sobolev and Besov spaces, the Stokes equation with the Robin boundary condition, and thermoelastic systems arising in shape memory alloys. No index is provided. (Annotation ©2008 Book News Inc. Portland, OR)

Asymptotic analysis and singularities; elliptic and parabolic PDEs and related problems.

Volume two of the two-volume set contains 24 papers presented at the July 2005 meeting held in Japan on nonlinear partial differential equations, dynamical systems, calculus of variations, and mathematical physics. Topics include analogies between superconductors and liquid crystals, stability analysis for a stripe solution in the Gierer-Meinhardt system, and concentration phenomena in the conformal Brezis-Nirenberg problem. No index is provided. Distributed in the U.S. by the American Mathematical Society. (Annotation ©2008 Book News Inc. Portland, OR)

Complex function theory, 2d ed.

A textbook for a one-semester graduate or undergraduate introductory course in the theory of functions of a complex variable for students of mathematics and related sciences. The goal is to provide a foundation for further study. Students are assumed to have a thorough grounding in basic real analysis. The first edition was published in 1994 as Notes on Complex Function Theory; the second corrects errors, flags exercises to which the solutions will appear later in the text, and appends background material that students can no longer be assumed to know. (Annotation ©2008 Book News Inc. Portland, OR)

Continuous symmetry; from Euclid to Klein.

This textbook defines and analyzes important classes of transformations of the plane, specifically isometries and similarities, and integrates transformations with the traditional geometry of lines, triangles, and circles. The second half of the book introduces Felix Klein's Erlanger program, uses the transformational approach to establish traditional theorems of plane geometry, and describes rosette groups, frieze groups, and wallpapers groups (Annotation ©2008 Book News Inc. Portland, OR)

Enhancing university mathematics; proceedings.

These proceedings of the May symposium held at the Korea Advanced Institute of Science and Technology (KAIST) address the needs of university-level mathematicians to develop effective ways of teaching undergraduate mathematics. The contributors, who are distinguished for their contributions to mathematics teaching, cover a wide range of topics at both broad and specific levels. Topics include the role of mathematics in general education, developing the enthusiasm of the best students, working through interactive geometry and multivariable calculus with the Internet, helping new faculty members, considering computer and human algebra, going international in teaching mathematics grad students to teach, inquiry-oriented mathematics instruction, using technology effectively, conducting research with undergrads, using real-world examples, choosing and using software, supporting blended learning, making math safe for social science majors, and using spreadsheets for real-life examples. (Annotation ©2008 Book News Inc. Portland, OR)

Finite fields and applications.

This work offers a brief and accessible introduction to the theory of finite fields and applications in combinatorics, algebraic coding, and cryptography. An appendix provides a brief review of basic algebraic concepts. Chapter exercises and hints are included. The book is accessible to undergraduate students with two years of standard college mathematics, and is suitable as a text for an upper division undergraduate course. The authors are affiliated with Pennsylvania State University. This book is part of a collection published jointly by the American Mathematical Society and the Mathematics Advanced Study Semesters (MASS) program as part of the Student Mathematical Library series. The books in the collection are based on lecture notes for advanced undergraduate topics courses taught at the MASS and/or Pennsylvania State University's summer program, Research Experiences for Undergraduates. (Annotation ©2008 Book News Inc. Portland, OR)

Frames for undergraduates.

The study of finite frames provides a strong setting for work on matrix analysis and the rudiments of finite-dimensional operator theory. Here, students learn the mathematical theory while simultaneously seeing how they can use operator theory to develop a body of results about frames. The authors start by reviewing linear algebra so less-seasoned students can catch up (and which more advanced students can skip), then move to finite-dimensional operator theory, finite frames, frames in the R plane, dual and orthogonal frames, frame operator decomposition, harmonic and group frames, and sampling theory. This also features student presentations on eigenspace decomposition, square roots of positive operators, polar decomposition, oblique projections and frames and Vandermonde determinants. The result is a concise and accessible treatment that serves well as a semester's text. (Annotation ©2008 Book News Inc. Portland, OR)

Geometric combinatorics.

This work compiles expository articles from a three-week program of lectures at the Institute for Advanced Study/Park City Mathematics Institute summer program on geometric combinatorics. Five lectures are presented for each topic. Topics include lattice points, polyhedra, and complexity; root systems and generalized associahedra; combinatorial differential topology and geometry; equivariant invariants and linear geometry; hyperplane arrangements; Poset topology; and convex polytopes. Most of the material is aimed at graduate students or researchers approaching the subject for the first time. Many of the chapters include exercises, and all include numerous examples. There is no subject index. (Annotation ©2008 Book News Inc. Portland, OR)

Geometry of conics.

Originally published in 2007, in Russian, by the Moscow Center for Continuous Mathematical Education, this book begins with coverage of the elementary properties of conics — material that can be approached with standard high school curriculum as background; the second chapter covers some auxiliary material not usually covered in high school but necessary for understanding the following, more complicated chapters on projective properties, and metric properties of conics. Some 50 exercises and problems (with solutions) offer support, along with approximately 100 carefully prepared figures. (Annotation ©2008 Book News Inc. Portland, OR)

In search of the Reimann zeros; strings, fractal membranes, and noncommutative spacetimes.

The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, remains one of the more important open problems of contemporary mathematics. This text proposes an approach towards proving the truth of the conjecture that is based in large part on analogies between the Riemann zeta function as expressed analytically by a functional equation and the dualities exhibited by string theories in theoretical physics. The argument makes no claims of solution and alternates between rigor and intuition. (Annotation ©2008 Book News Inc. Portland, OR)

Index theorem I.

Furuta presents the second proof of the Atiyah-Singer index theorem, avoiding the use of pseudo-differential operators, and defines the index for operators of Dirac type not only on closed submanifolds, but also on open submanifolds with certain conditions at infinity for the operator. The monograph develops examples of the localization of the index, a localization of eigenfunctions of the Laplace operator, and characteristic classes. Originally published in Japanese by Iwanami Shoten in 1999. (Annotation ©2008 Book News Inc. Portland, OR)

An initiation to logarithmic Sobolev inequalities.

This work introduces Sobolev inequalities and discusses important applications to mathematical statistical physics. The book begins by reviewing the necessary background on selfadjoint operators, semigroups, Kolmogorov diffusion processes, and solutions of stochastic differential equations. This is followed by a chapter on log Sobolev inequalities with an application to a strong ergodicity theorem for Kolmogorov diffusion processes. The remaining two chapters consider the general setting for Gibbs measures, and the application of the Ising model to log Sobolev inequalities to show the stabilization of the Glauber-Langevin dynamic. Appendixes cover Markovian kernels, bounded real measures, and the topology of weak convergence. The book was originally published in French in 1999 by Société Mathématique de France. Author information is not given. There is no subject index. (Annotation ©2008 Book News Inc. Portland, OR)

The interface between convex geometry and harmonic analysis.

Central to geometry and particularly useful in applications to other areas of mathematics as well, this aspect of study of convex bodies and the geometry of sections has been advanced by methods from Fourier analysis, the idea being to express certain properties of bodies in terms of the Fourier transform and then to use methods of Fourier analysis to solve geometric problems. Developed from material presented at the conference of the same name held July and August 2006 at Kansas State U. in Manhattan, Kansas, this is written in the form of lectures accessible to graduate students and contains information on the newest advances. It covers hyper-plane sections of lp-balls, volume and the Fourier transform, intersection bodies, the Busemann-Petty problem, projections and the Fourier transform, intersection bodies and Lp-spaces, between polar projection bodies and intersection bodies, and open problems. (Annotation ©2008 Book News Inc. Portland, OR)

Interpolation theory and applications; proceedings.

Interpolation theory considers a function f, defined on a regular grid N. The problem is to find f in a continuum that includes N. This collection of 23 papers represent the March 29-31, 2006 conference on interpolation theory and applications, dedicated to mathematician Michael Cwikel (Technion-Israel Institute of Technology), in recognition of his many contributions to this field. A synopsis of Cwikel's career by Milman (Florida Atlantic U.) and list of his publications opens the volume. Cwikel contributes a proof of a long-standing question regarding complex interpolation spaces. Other topics treated include classical analysis, harmonic analysis, function spaces, image processing, and the geometry of Banach spaces. Photos of some of the participants are included, but not an index. (Annotation ©2008 Book News Inc. Portland, OR)

Invitation to ergodic theory.

Silva introduces ergodic theory, or measurable dynamics, to upper-level undergraduate mathematics students who have completed a basic course in real analysis. He does not assume knowledge of measure theory, and develops it as needed. Among the topics are recurrence, ergodicity, the ergodic theorem, and mixing. (Annotation ©2008 Book News Inc. Portland, OR)

Lectures on analytic differential equations.

This graduate-level textbook includes features of research monographs and surveys of recent research on the analysis and geometry of differential equations in the real and complex domain. It covers normal forms and desingularization, including analytic differential equations in the complex domain, holomorphic foliations and their singularities, finitely generated groups of conformal germs, and desingularlization in the plane; singular points of planar analytic vector fields, including planar vector fields with characteristic trajectories, holonomy and first integrals, and complex separatrices of holomorphic foliations; the local and global theory of linear systems, including Riemann-Hilbert problems; functional moduli of analytic classifications of resonant germs and their applications, including complex saddles and saddle-nodes; and global properties of complex polynomial foliations. The authors provide a crash course on functions of several complex variables and elements of the theory of Riemann spaces. (Annotation ©2008 Book News Inc. Portland, OR)

Lectures on counterexamples in several complex variables.

This work presents lectures from a graduate course at Princeton in 1982-83, collecting some of the counterexamples in the several-complex-variable theory which were scattered throughout the literature. The first part of the book introduces some of the basic concepts of the theory, in particular those needed later for the examples. The counterexamples cover topics such as the Levi problem, plurisubharmonic functions, CR geometry, and function theory. The book can serve as a supplement to a graduate course on several complex variables. There is no subject index. (Annotation ©2008 Book News Inc. Portland, OR)