American Mathematical Society

1001 problems in classical number theory.

There really are 1001 problems in classical number theory here, and each one leads to the next so readers can progress at their own speed. As they read they will be enticed into trying just one more, succeeding, and moving on to the next, Ideal for self-study or for a year's worth of exercises, this starts with key elements from the theory such as argument, inequalities, divisibility, prime numbers, congruence, arithmetic functions, diophantine equations, and classifications of real numbers. The authors then work through the problems together with readers following basically the sequence in the text and providing solutions. The result is quite accessible and the innovative approach could serve as a model for a lively series. (Annotation ©2007 Book News Inc. Portland, OR)

Advances in logic; proceedings.

In these proceedings from the October 2004 conference, participants describe their work in diverse fields of logic that contain significant new results that are accessible to readers with a general background in logic. This includes a problem list compile by the speakers that reflects some of the most important questions in various areas of logic, and the presenters cover such topics as a stationary-tower-free proof of the derives model theorem, a proof of an absoluteness theorem, a simple inductive measure analysis of cardinals under the axiom of determinacy, the complexity of index set and Ehrenfeucht theories, computable structures in familiar classes, the classes of separating sets, voting rules for infinite sets and Boolean algebras, "very mad families," Borel boundedness and the lattice surrounding property, and Steinhous sets and Jackson sets. (Annotation ©2007 Book News Inc. Portland, OR)

Algebraic and geometric combinatorics; proceedings.

These proceedings from the August 2005 conference include both original research and survey articles, focusing on interactions of combinatories with other branches of mathematics, such as commutative algebra, algebraic geometry, convex and discrete geometry, enumerative geometry, and topology of complexes and partially ordered sets. Aimed at both researchers and graduate students interested in various aspects of combinatorial theories, topics here include combinatories of polytopes, lattice polytopes, triangulations and subdivisions, Cohen-Macaulay cell complexes, monomial ideals, the geometry of toric surfaces, groupoids in combinatories, Kazhdan-Lusztig combinatories and graph colorings. The editors include a full list of participants. (Annotation ©2007 Book News Inc. Portland, OR)

Algebraic geometry; proceedings.

These proceedings of July 2004 include articles examining interactions between algebraic geometry and other branches of mathematics, as so they should in honor of the work of Dolgachev. Topics include algebraic curve theory, algebraic surface theory, moduli space, automorphic forms, Mordell-Weil lattices and automorphisms of hyperkahler manifolds. Specific papers address holomorphic Eisenstein series with Jacobian twists, Hessians and the moduli surface of cubic surfaces, covariants of a pane quartic associated to its even theta characteristics, a partial solution to Kollar's conjecture, invariants of quartic plane curves as automorphic forms, correspondences of a K3 surface with itself, automorphisms of hyperkahler manifolds in the view of topological entropy, classical Kummer surfaces and Mordell-Weil lattices, and Newmeier lattices and K3 groups. (Annotation ©2007 Book News Inc. Portland, OR)

The Beilinson complex and canonical rings of irregular surfaces.

Canonaco extends Beilinson's theorem describing the bounded derived category of coherent sheaves on Pⁿ to weighted projective spaces, considering not the usual category of coherent sheaves on P(w), but a suitable category of graded coherent sheaves obtained by endowing P(w) with a natural graded structure sheaf. He then applies the weighted version of the theorem to prove a structure theorem for good birational weighted canonical projections of surfaces of general type. (Annotation ©2007 Book News Inc. Portland, OR)

Complex geometry and related fields; proceedings.

These proceedings from the 2004 conference describe significant advances in differential and algebraic geometry and their connections to number theory and mathematical physics. Topics include optimal control of the Lieuville equation, real structures on torus bundles and their deformations, , representations of finite Lie algebras and geometry of reductive lie algebras, multi-parameter cells of finite Coxeter groups, rigidity problems in Cauchy-Riemann analysis, periods of automorphic forms, problems with Hamiltonian line graphs, localization and string quality, the second main theorem with hypersurfaces over function fields, presentations for finite complex reduction groups, perspectives on geometric analysis, automorphisms of K3 surfaces and vector bundles on certain surfaces without divisors. (Annotation ©2007 Book News Inc. Portland, OR)

Combinatorial group theory, discrete groups, and number theory; proceedings.

These proceedings for the December 2004 conference held in honor of Gerhard Rosenberger include materials from the AMS session on infinite groups held October 2005, and the title topics are joined by those on ring theory as well as contributions to noncommutative algebraic cryptography. Papers cover such subjects as outer automorphism groups of certain orientable Seifert three-manifold groups, a proposed public key cryptosystem using the modular group, normal subgroups of themodular group and other Hecke groups, unions of varieties and quasi-varieties, context-free irreducible word problems in groups, informative words and discreteness, using group theory for knowledge representation and discovery, torsion in maximal arithmetic Fuchsian groups, density of test elements in finite Abelian groups and the Rosenberg "monster." (Annotation ©2007 Book News Inc. Portland, OR)

Control and nonlinearity.

This presents methods for graduate students and mathematicians in control theory to help them study the controllability and stabilization of nonlinear control systems in finite and infinite dimensions. To reinforce one of the main points, of the examples involve non-linearities essential to understanding controllability or stabilization. This covers finite-dimensional linear control systems, linear partial differential equations, controllability of nonlinear systems in finite dimensions, linearized control systems and fixed-point methods, iterated Lie brackets, return methods, quasi-state deformation, power series expansion, Schrödinger equations, linear control systems in finite dimensions and applications to nonlinear control systems, stabilization of nonlinear control systems in finite dimensions, feedback design tools, and applications to some partial differential equations. (Annotation ©2007 Book News Inc. Portland, OR)

The conceptual foundations of quantum mechanics.

Originally published in 1971, this work by the late Eisenbud (then of the department of physics at the State U. of New York in Stony Brook) aimed to provide a conceptual understanding of the foundations of quantum physics by downplaying formal mathematical calculations in favor of investigation of physical meanings. Over the course of nine chapters, he analyzed the physical meaning and conceptual consequences of the Heisenberg principle, paying particular attention to the incompatibility of pairs of observables. He also explored the effects of the concept of incompatibility on the meanings of "measurement," "property," "state," and "indeterminism." He discussed the existence and significance of probability amplitudes. As mentioned above, the work is only intended to provide foundations and should not be considered an introduction to quantum physical theory itself. (Annotation ©2007 Book News Inc. Portland, OR)

Control methods in PDE-dynamical systems; proceedings.

Sixteen papers from the July 2005 conference study both controlled partial differential equation (PDE) systems and the asymptotic long- time behavior of PDE-mixed problems. The opening paper presents results on the asymptotic stabilization of systems of conservation laws by controls acting at a single boundary point. An equally long paper develops the dynamics of a semilinear wave equation with nonlinear interior/boundary dissipation. Other topics include variational principles for finite dimensional initial value problems, optimality conditions for solutions to hyperbolic balance laws, microscale sensitivity in moving boundary problems for the thin-film equation. No subject index is provided. (Annotation ©2007 Book News Inc. Portland, OR)

Crossed products of C*-algebras.

Work in crossed products has proven to be extremely rich and valuable to such disparate fields as noncommutative geometry and mathematical physics. Here Williams (mathematics, Dartmouth U.) provides a self-contained and accessible text for graduate students in first courses on operator algebras. He covers locally compact groups, dynamical systems and crossed products, special cases and basic constructions, imprimitivity theorems, induced representations and induced ideals, orbits and quasi-orbits, properties of crossed products, ideal structure and the proof of the GRS theorem. Several of his appendices also offer independent interest, covering amenable groups, Banach algebra, bundles of C*-algebras, groups, representations of C*-algebras, direct integrals, Effros's ideal center decomposition, the Fell topology. (Annotation ©2007 Book News Inc. Portland, OR)

Discrete mathematics.

Aigner (mathematics, First U. of Berlin) writes for computer scientists and mathematicians with an interest in the main ideas behind discrete mathematics, especially algorithms. He covers counting, including the fundamentals, summation, generating functions, counting patterns, asymototic analysis; graphs and algorithms, including trees, matchings and networks, searching and sorting and general optimization methods; and algebraic systems, including Boolean algebras, modular arithmetic, coding, cryptography and linear optimization. For this English-language edition Aigner has added the chapters on counting patterns and symmetries and on coding. With its now-600 exercises, all with hints and about half with solutions, and section bibliographies, college students with an understanding of linear algebra and undergraduate calculus will also find this text helpful. (Annotation ©2007 Book News Inc. Portland, OR)

Fredholm operators and Einstein metrics on conformally compact manifolds.

Building on the foundation set by Fefferman and Graham in their approach to the study of local invariants of conformal structures, Lee gives an elementary of the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities sufficient close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. Lee bases his proof on an elementary derivation of sharp Fredholm theorems for self-adjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds. His topics include Mobius coordinates, function spaces, elliptic operators, the analysis of hyperbolic space, Fredholm theorems, Laplace operators and Einstein metrics. (Annotation ©2007 Book News Inc. Portland, OR)

Harmonic analysis, partial differential equations, and related topics; proceedings.

These proceedings of the October 2005 seminar includes the papers that comprise the scientific program, and include such fields as partial differential equations, harmonic analysis and Fourier analysis. The articles illustrate the interaction among the fields and individual topics include Painlevé removeability, pseudo-differential operators, certain weights, nonlinear Schröeder equations, singular integrals, the wave equation, the Benjamine-Ono equation, quasi-geostrophic equations, quasi-conformal mappings, integral inclusions, Bellman function methods, weighted gradient estimates, Hankel operators and dynamic optimization problems. One of the most significant findings of these papers are the successful applications of techniques and ideas across fields. This includes the full program from the seminar. (Annotation ©2007 Book News Inc. Portland, OR)

The interaction of analysis and geometry; proceedings.

These proceedings of the August-September 2003 conference include articles based on papers presented in honor of Yurii Reshetnyak. In general, the articles address the geometry of spaces with bounded curvature in the manner of Alexandrov, quasiconformal mappings and mappings with bounded distortion (or "quasiregular mappings"), nonlinear potential theory, Sobolev spaces, spaces with fractional and generalized smoothness, variational problems, and new research in these fields of study. Most articles relate to Reshetnyak's own interests, which include geometry in the large sense, quasiconformal analysis, Sobolev space, potential theory and variational calculus. (Annotation ©2007 Book News Inc. Portland, OR)

Linear algebra in action.

Dym describes how linear algebra, perhaps more than any other subject, permeates mathematics. He not only explains how and why that is so, he also demonstrates linear algebra's nature through a wide variety of applications, covering vector spaces, Gaussian eliminations, eigenvalues and eigenvectors, determinants, calculating Jordan forms, normal linear spaces, inner product spaces and orthogonality, matrices, singular values and related inequalities, pseudoinverses, difference equations and differential operations, vector related functions, external problems, matrix values holomorphic functions, realization theory, eigenvalue locator problems, zero location problems, convexity and matrices with nonnegative spaces. (Annotation ©2007 Book News Inc. Portland, OR)

Modular forms; a computational approach.

Stein (mathematics, U. of Washington) fills the gap in the literature on classical modular forms with this unique approach that centers on computation throughout, defining what modular forms are and showing in detail how one can compute everything about them in practice. He uses examples from his own free software package, making this a suitable text for beyond the introductory level. Stein appeals to graduates, advanced undergraduates and non-specialists in number theory as be describes the modular forms of weights and levels, Dirichlet characters, Eisenstein series and Bernoulli numbers, dimensions formulas, linear algebra, general modular symbols, computing with newforms, and computing periods. He includes solutions to selected exercises. (Annotation ©2007 Book News Inc. Portland, OR)

Moduli spaces and arithmetic geometry; proceedings.

In these proceedings from the September 2004 conference, which is dedicated to Professor Masaki Maruyama, contributors describe their research in algebraic geometry and number theory. Their topics include moduli spaces of twisted sheaves on a projective variety, integral Hodge classes on uni-ruled or Calabi-Yau threefolds, birational geometry of symplectic resolutions of nilpotent orbits, moduli stacks of second-rank Giesker bundles with a fixed determinate on a nodal curve, vector bundles on curves and theta functions, Abelian varieties with bounded modular height, the moduli of regular holonomic Dx-modules with natural parabolic stability, cohomology groups of stable quasi-Abelian degenerations, semi-stable extensions on arithmetic surfaces, cusp form motives, polarized K3 surfaces, rigid geometry and applications, and moduli of stable parabolic conventions with Riemann-Hilbert correspondence and other features. Distributed in the US by the American Mathematical Society. (Annotation ©2007 Book News Inc. Portland, OR)

Nonlinear equations and spectral theory.

Dedicated to the memory of mathematician O. A. Ladyzhenskaya, this volume contains ten articles (mainly devoted to boundary value problems for partial differential equations and to spectral problems for differential operators) written by her students and colleagues. Specific topics include quasireverse Holder inequalities in parabolic metrics and their applications, Weyl asymtotics of the spectrum of the Maxwell operator with non-smooth coefficients in Lipschitz domains, semi-classical pseudo-differential operators with discontinuous symbols and their applications to the problems of statistical physics, complete integrability in quantum mechanics, geometric evolution equations preserving convexity, spectral properties of elliptic problems in domains with cylindrical ends, weak solutions in the Cauchy problem for the Navier-Stokes equations satisfying the local energy inequality, Schauder estimates for the evolutionary generalized Stokes problems, homogenization of a periodic parabolic Cauchy problem and boundary estimates for solutions of elliptic and parabolic equations with discontinuous nonlinearities. (Annotation ©2007 Book News Inc. Portland, OR)

P-adic analysis compared with real.

The series presents lecture notes for advanced undergraduate courses taught by the American Mathematical Society and the Mathematics Advanced Study Semesters (MASS) program. Katok gave her MASS course in the fall of 2000; the notes were published in MASS Selecta and in a Russian version, but she has revised and expanded them for this volume. The many exercises are included, along with the homework assignments. (Annotation ©2007 Book News Inc. Portland, OR)