American Mathematical Society
Advances in string theory; the First Sowers Workshop in Theoretical Physics; proceedings.
String theory may be exactly that, a theory, but it has nevertheless strongly influenced thinking in such fields of research as high energy and hadronic physics, gravitation, cosmology, mathematical physics and even condensed matter physics. Editors Sharpe (physics, Virginia Polytechnic Institute and State U.), Greenspoon (mathematics, U. of Michigan) and their contributors cover a wide range of subjects, including Gromov-Witten theory, mirror symmetry in complex and symplectic geometry, and important ramifications in enumerative geometry, with specific papers addressing strong coupling, brane-induced gravity, heterotic geometry influxes, pure Yang-Mills theory, black hole microstates, gravitational singularity, recent developments in heterotic compactifications, instantons and torsion curves. Includes a list of participants. A co-publication of the AMS and International Press. (Annotation ©2009 Book News Inc. Portland, OR)
Algebra; a graduate course.
This mathematics textbook for graduate students covers the fundamentals of abstract algebra including fields and Galois theory, algebraic number theory, algebraic geometry and groups, rings and modules. Isaacs (mathematics, U. of Wisconsin- Madison) also includes coverage of rarer subjects in this field, including transcendental field extensions, modules over Dedekind domains and artinian rings. Unlike similar textbooks, this volume steers away from chapter-end problems by including full details of all proofs as problems are presented. (Annotation ©2009 Book News Inc. Portland, OR)
Class field theory, 2d ed. (reprint, 1967)
Artin (1898-1962) and Tate presented a seminar on the theory at Princeton University in 1951-52, and assembled for publication their lecture notes on the parts dealing with global class field theory, and the abstract theory of class formations and Wiel groups. Tate has slightly revised the treatise, added some footnotes and historical comments in order to compensate for the lack of references and attribution of credit, and enlarged the discussion on Wiel groups by a few pages. There is no index. The first edition was published in 1967 by W. A. Benjamin, New York. (Annotation ©2009 Book News Inc. Portland, OR)
Combinatorial geometry and its algorithmic applications; the Alcala lectures.
The two mathematicians delivered the lectures at Alcalá, Spain in August-September 2006, drawing mostly on their previous work, both separate and collaborative. They survey a number of topics within combinatorial geometry for professional and student mathematicians. Among them are the evolution of the basic theory of arrangements of surfaces, incidences and their relatives from Szemerédi and Trotter to cutting lenses, repeated patterns and pattern recognition, and geometric coloring problems. (Annotation ©2009 Book News Inc. Portland, OR)
Computational geometry of positive definite quadratic forms; polyhedral reduction theories, algorithms, and applications.
Schürmann (mathematics, U. of Magdeburg, Germany) discusses classical questions on quadratic forms and their connections to lattice sphere packing and covering problems. Designed for advanced mathematics and physics students, this book explores the theories of Minkowski and Voronoi in unprecedented depth, revealing new applications to the properties of exceptional structures such as root lattices and the Leech lattice. Use of computer software applications to solve and explore these problems is encouraged throughout the text. (Annotation ©2009 Book News Inc. Portland, OR)
Combinatorics on words; Christoffel words and repetitions in words.
Notes from two series of five two-hour lectures have been expanded into articles and augmented with over 100 exercises. They were delivered in March 2007 at the Centre de recherches mathmatiques (CRM) in Montreal. Christophe Reutenaur gave the first five lectures, on Christoffel words, which are finitary versions of Sturmian sequences. The others, by Berstel, begin with an exposition of the numerous properties, applications, and interpretations of the famous Thue-Morse word, then explore Thue's construction of a square-free word, a linear-time algorithm for findings squares in words, and other problems. (Annotation ©2009 Book News Inc. Portland, OR)
Discrete differential geometry; integrable structure.
This textbook on discrete differential geometry is aimed at mathematics and physics students who have completed courses in calculus and linear algebra and need to master associated concepts in geometry processing, numerical simulations and applications for computer graphics and animation. Bobenko (mathematics, Technische U. Berlin) and Suris (mathematics, Zentrum Mathematik Technische U. München) explain the nets of classical differential geometry before moving on to discretization principles for multidimensional nets and nets in quadrics, special classes of discrete surfaces, approximation and discrete complex analysis. Projective geometry, Lie geometry and incidence theorems are also discussed. (Annotation ©2009 Book News Inc. Portland, OR)
Ergodic theory, groups, and geometry; proceedings.
The study of group actions on manifolds is the meeting ground of a variety of mathematical areas. In particular, interesting geometric insights can be obtained by applying measure theoretic techniques. The lecture notes collected here provide an introduction to some of the important methods, major developments, and open problems in the subject. They are slightly expanded from lectures given by the author at a 1998 CBMS Conference held at the University of Minnesota. The main text presents a perspective on the field as it was 10 years ago; comments after the notes of each lecture provide suggestions for further reading, including references to recent developments. Some topics examined are superrigidity and first applications, locally homogeneous spaces, and orbit equivalence. An appendix of background material is included. (Annotation ©2009 Book News Inc. Portland, OR)
Finite dimensional algebras and quantum groups.
The quantum groups investigated in this book are quantum enveloping algebras defined by their Drinfeld-Jimbo presentation once a generalized Cartan matrix is specified. The book consists of 14 chapters arranged in five parts on quivers and their representations, quantized algebras, representations of symmetric groups, Ringel-Hall algebras, and the BLM algebra. These are complemented by an introductory chapter on two realizations of Cartan matrices, and three appendices on varieties and affine algebraic groups, quantum linear groups through coordinate algebras, and quasi-hereditary and cellular algebras. Chapters and appendices include 8-10 pages of exercises and notes, ranging from routine sketches of proofs to recent results from current literature. The book is suitable for advanced graduate students. (Annotation ©2009 Book News Inc. Portland, OR)
Fourth Summer School in Analysis and Mathematical Physics; topics in spectral theory and quantum mechanics; proceedings.
Drawn from their lectures given at the May 2005 summer school, these three expository articles connect the basic knowledge of a graduate student in mathematics with isoperimetric inequalities for eigenvalues of the Laplace operator, random Schrödinger operators, and the stability of matter in terms of quantum mechanics and quantum electrodynamics. The authors (Rafael Benguria, Peter Hislop, and Elliott Lieb) make every effort to relate these topics to the development of spectral theory and provide excellent examples. The final result is a professional reference as well as a solid supplemental text for courses in quantum mechanics or spectral theory. Co-published with Sociedad Matematica Mexicana. (Annotation ©2009 Book News Inc. Portland, OR)
From Hodge theory to integrability and TQFT; tt*-geometry; proceedings.
By the time Cecotti, Vafa and their coauthors published their masterwork on the geometry of topological field theory in 1991, quantum field theory and string theory had already had an enormous impact on geometry. Drawn from discussions held at the U. of Augsburg in May 2007, these papers include an overview of the underlying geometric functions and structures along with new research about "tt*" geometry and its role in singularity theory, Hodge theory, integrable systems, matrix models, and Hurwitz spaces. Specific topics include the universal unfolding of Laurent polynomials and tt* structures, applications from primitive forms to Frobenius manifolds, twistor structures, tt* geometry and singularity theory, differential aspects of tt* equations, aspects of Hodge's theoretical mirror symmetry, applicability to the Neumann system, dimensional gauge series and quantum integrable systems, Hurwitz numbers and matrix models in enumerative geometry, and topological string functions. (Annotation ©2009 Book News Inc. Portland, OR)
Function theory; interpolation and corona problems.
Sawyer's (mathematics and statistics, McMaster U.) contains lecture notes delivered during a graduate course on function theory at the Fields Institute in March 2008. It takes readers from Lennart Carleson's first deep results on interpolation and corona problems in the unit disk to modern analogues in the disk and ball. Specific topics include Toeplitz and Hankel operators, Hilbert function spaces and Nevanlinna-Pick kernels, functional analysis, and function theory on the disk. While intended for relative novices in the field, the author notes that readers should be acquainted with basic and complex analysis and the theory of the Poisson integral in the unit disk. (Annotation ©2009 Book News Inc. Portland, OR)
Geometric analysis on symmetric spaces, 2d ed.
This book gives an account of geometric analysis on Riemannian symmetric spaces and its relationship to the representation theory of Lie groups. After discussing the theory of Radon transforms on symmetric spaces, inversion formulas, and range theorems, the book examines applications to invariant differential equations on symmetric spaces, existence theorems, and explicit solution formulas, particularly potential theory and wave equations. It concludes with a chapter on eigenspace representations. Chapter exercises and solutions are included. This second edition further develops the theory of the Fourier transform and the horocycle transform on X, also taking into account tools developed by Eguchi for the Schwartz space S(X). The author received the 1988 Steele Prize for his earlier publications. The book can be used in advanced graduate courses in modern integral geometry, analysis on Lie groups, and representation theory of Lie groups. (Annotation ©2009 Book News Inc. Portland, OR)
The geometry of Heisenberg groups; with applications in signal theory, optics, quantization, and field quantization.
Written for graduate students of both fields, this volume examines and connects the mathematics and physics of Heisenberg groups. Ten chapters discuss the skew field of quaternions; elements of the geometry of S3, Hopf bundles, and spin representations; internal variables of singularity free vector fields in a Euclidean space; isomorphism classes, Chern classes, and homotopy classes of singularity free vector fields in 3-space; Heisenberg algebras, Heisenberg groups, Minkowski metrics, Jordan algebras, and special linear groups; the Heisenbreg group and natural C*-algebras of a vector field in 3-space; the Schrödinger representation and the metaplectic representation; the Heisenberg group as a basic geometric background of signal analysis and geometric optics; quantization of quadratic polynomials; and field theoretic Weyl quantization of a vector field in 3-space. Also included is an appendix on thermodynamics, geometry, and the Heisenberg group. Minimal prerequisites include background in linear algebra and elementary differential geometry. (Annotation ©2009 Book News Inc. Portland, OR)
Graduate algebra; noncommutative view.
Volume 73 in the series presents the commutative view, and this textbook continues it, starting with chapter 13. It grew out of a course introducing complex presentations of finite groups through the structure of group algebras and their modules. It is primarily concerned with casting as much of noncommutative theory as possible in terms of representations into matrix algebras. A second theme is describing algebraic structures in terms of generators and relations, leading to discussions of free structures, growth, word problems and Zelmanov's solution of the Restricted Burnside Problem. (Annotation ©2009 Book News Inc. Portland, OR)
Lessons in geometry; I. plane geometry. (CD-ROM included)
Saul presents the first English translation of this textbook in the tradition of Euclidean synthetic geometry, written in 1898 and revised twelve times by the French scholar Hadamard. The translation of this first volume — on plane geometry — is based on a 1947 reprint of the thirteenth edition. The text contains exercises and may be of particular interest to advanced high school students and undergraduates in mathematics seeking a more sophisticated understanding of geometry. The accompanying CDROM contains solutions, using Texas Instruments' TI-Nspire software, to select problems. (Annotation ©2009 Book News Inc. Portland, OR)
Markov chains and mixing times; with a chapter on coupling from the past by James G. Propp and David B. Wilson.
This introduction to the theory of Markov chains describes a modern approach to determining the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the state space. Part I focuses on techniques, with accessible examples. Part II covers more sophisticated techniques and presents several detailed case studies of particular families of chains. The authors develop tools for estimating convergence times, including coupling, strong stationary times, and spectral methods. Whenever possible, probabilistic methods are emphasized. The book includes many examples and provides brief introductions to some central models of statistical mechanics. Also provided are accounts of random walks on networks, including hitting and cover times, and analyses of several methods of shuffling cards. Chapter ending notes give references to original papers and provide suggestions for further reading. As a prerequisite, the authors assume a modest understanding of probability theory and linear algebra at an undergraduate level. Levin is affiliate with the University of Oregon. (Annotation ©2009 Book News Inc. Portland, OR)
Operator algebras and geometry.
Originally published in Japanese for a Surveys in Geometry workshop held in Tokyo, Japan, in 1998, this translation offers an overview of operator algebra theory. Intended for topologists and other mathematicians specializing in geometry, the translation into English by Moriyoshi and Natsume (for whom no background information is given) is intended for readers who do not have a thorough knowledge of functional analysis. The book includes explanations of basic tools used in noncommunicative geometry and applications to the Atiyah- Singer type index theorems. (Annotation ©2009 Book News Inc. Portland, OR)
Ordering braids.
In 2002, the same gang of four authors wrote Why Are Braids Orderable and published it in a French Mathematical Society series. This is a sequel, they say, not a second edition: though it contains most of the previous information, it has been totally rewritten and scads of new information has been added. The question in the original title has shifted somewhat as well: They still do not know what makes the braid groups so special that they enjoy an ordering so easy to describe, so challenging to construct, and with such subtle properties. So rather than asking why they can be ordered, they are satisfied with describing how. Readers are assumed to have some background in group theory and topology. (Annotation ©2009 Book News Inc. Portland, OR)
p-adic geometry; lectures from the 2007 Arizona Winter School.
Taken from four lectures held at the annual school in March 2007 at the U. of Arizona, these lecture notes are accessible to both graduate students and researchers interested in learning about the techniques of p-adic geometry. This field, which has provided tools to number theory, algebraic geometry, and the theory of automorphic representations, has produced significant research in recent years as these lecture show. Topics include non-archimedean analytical geometry, approaches to non-archimedean geometry, the p-adic upper half plane, Berkovich analytic spaces and non-archimedean potential theory on curves, and p-adic cohomology from theory to practice. Each lecture includes full references. (Annotation ©2009 Book News Inc. Portland, OR)
