American Mathematical Society
Logic's lost genius; the life of Gerhard Gentzen.
Under the nazis there was "German science," "German mathematics and, of course, "German logic." Then there was Gentzen, the founder of modern structural proof theory. Although his methods, rules and structures have lead to verification programs essential to computer science, and his work on natural deduction, the sequent calculus and ordinal proof theory are still considered advanced, his life eventually ran contrary to the passions of his nation. Although he did not seem to be offended by nazism personally, and enjoyed some measure of protection from powerful nazi friends, his thoughts were generally dedicated to his work. Menlzer-Trott combines a close understanding of that work with extensive research into primary resources such as family archives to give a fully-fleshed portrait of a genius caught in fire, whose death from hunger typhus in a Prague prison was an illogical obscenity. (Annotation ©2008 Book News Inc. Portland, OR)
Mathematics as metaphor; selected essays of Yuri I. Manin.
The emininent physicist and mathematician Freeman Dyson provides a foreword to this collection of essays, noting that "Some mathematicians are birds, others are frogs. Birds [like Manin] fly high in the air and survey broad vistas...." Comprising about a dozen papers published during the last three decades, this volume is devoted to Manin's philosophical thoughts on what's important in mathematics, with excursions into history and into how mathematicians think. Though identified as non-technical (a selection of Manin's technical papers was published in 1996), the papers are for mathematically sophisticated readers, but not necessarily mathematicians. The book would lend itself well to a college-level seminar for physicists, biologists, engineers, and others, with selections grouped into sections on mathematics as metaphor; mathematics and physics; and language, consciousness, and book reviews. The final selection is an interview with Manin titled "Good proofs are proofs that make us wiser." (Annotation ©2008 Book News Inc. Portland, OR)
Moscow seminar on mathematical physics, 2.
This work contains proceedings of seminars of the Mathematical Physics Group of the Institute for Theoretical and Experimental Physics. Papers in the collection, 12 in all, are devoted to various mathematical topics that strongly influenced modern physics. Among these topics are cohomology and representations of infinite Lie algebras and superalgebras, Hitchin and Knizhnik-Zamolodchikov-Bernard systems, and the theory of D-modules. Some specific paper topics include combinatorics and geometry of higher level Weyl modules, the Cosh-Gordon equation and quasi-Fuchsian group syzygies of highest weight orbits, and fermionic Fock space. The book is intended for graduate students and research mathematicians working in algebraic geometry, representation theory, and mathematical physics. (Annotation ©2008 Book News Inc. Portland, OR)
Noncommutative geometry, quantum fields and motives.
Connes (College de France and Vanderbilt) and Marcolli (Max Planck Institute and Florida State) explain the relevance of noncommutative geometry in dealing with the construction of a theory of quantum gravity and the Riemann hypothesis. Also appropriate for physicists, the advanced mathematics textbook describes the Standard Model of elementary particles, a spectral realization of the zeros of the Riemann zeta function, quantum statistical mechanical systems, and a cohomological Lefschetz trace formula. (Annotation ©2008 Book News Inc. Portland, OR)
Partially hyperbolic dynamics, laminations, and Teichmüller flow.
Researchers of dynamical systems who specialize in partially hyperbolic dynamics and Teichmüller flow participated in a workshop at the Institute in January 2006. The approach of the gathering, and of the 15 papers here that emerged from it, emphasized the theory of laminations and foliations, dynamical hyperbolicity, and ergodic theory. No index is provided. (Annotation ©2008 Book News Inc. Portland, OR)
Perspectives in nonlinear partial differential equations; in honor of Haïm Brezis.
Honoring Brezis (U. P. and M. Curie, Paris) on the occasion of his 60th birthday at the 2004 Paris meeting of the American Mathematical Society, the volume begins with a speech — untranslated from the French — delivered by this major influence on nonlinear analysis and partial differential equations. Berestycki (U. Paul Cézanne, Marseille, France) and other international mathematicians introduce this volume of 23 papers on the state of the art in this field. Contributors present new perspectives on multiple aspects of topics impacted by Brezis including nonlinear Schrödinger equations, contact form geometry, models of image processing, Sobolev maps, and game theory interpretations of nonlinear problems. (Annotation ©2008 Book News Inc. Portland, OR)
Prediction and discovery; proceedings.
For research mathematicians interested in the latest theoretical developments relating to machine learning and for scientists and engineers interested in real-world applications, Verducci (Ohio State U., Columbia) introduces 14 papers from this conference held June 25-29, 2006. Intended as a bridge between the statistics and computer science communities, the conference addressed the themes of support vector machine (SVM) learning, boosting and assembly methods, random networks, and models for approximate inference. Application areas discussed include new SVM models for bank fraud detection, hyperclique methods for protein detection, and multi-level spatial models for medical diagnosis. The volume is not indexed. (Annotation ©2008 Book News Inc. Portland, OR)
Pseudo-differential operators; partial differential equations and time-frequency analysis.
Based on five mini-courses and 15 of the lectures given at the workshop held in 2006 in Toronto by the Fields Institute, these papers address partial differential equations, geometric analysis, Fourier integral operators, localization operators, Gabor transforms, wavelet transforms, Rihaczek transforms and time-frequency analysis. Papers also cover the mathematical underpinnings, applications and ramification of the relatively new Stockwell transform, a combination of the Gabor and wavelet transforms. Specific topics include Hörmander operators and non-holonomic geometry, Weyl transforms and the inverse of the sub-Laplacian on the Heisenberg group, corner operators and applications to elliptic complexes, semilinear pseudo-differential equations and traveling waves, trace ideals for Fourier integral operators with non-smooth symbols, the S-transform and why to use it, inversion formulas for two-dimensional Stockwell transforms, Shannon-type sampling theorems on the Heisenberg group, and a unified point of view on time-frequency representatives and pseudo-differential operators. Directed at graduate students and post-doctoral researchers. (Annotation ©2008 Book News Inc. Portland, OR)
The Ricci flow; techniques and applications, pt.2: Analytic aspects.
Describing the effects of the depth and breadth of this flow method on the understanding the structure of manifolds, the authors begin with the tenth chapter in this series, focusing on weak maximum principles for scalars, tensors and systems, then move to closed manifolds and positive curvature, weak and strong maximum principles on non-compact manifolds, qualitative behaviors of classes of solutions, differential Harnack estimates of LYH types, and Perelman's differential Harnack estimate. Appendices give an overview of aspects of Ricci flow, aspects of geometric analysis related to Ricci flow, and tensor calculus on the frame bundle. They provide a number of worked examples and exercises along with some very interesting quotations from unlikely sources. The result is lively and very accessible to professionals and advanced students. (Annotation ©2008 Book News Inc. Portland, OR)
Roots to research; a vertical development of mathematical problems.
This volume provides a source for the math needed to understand the emergence and evolution of five contemporary problems whose development can be traced through elementary, high school, college, and university level math: the four numbers problem, rational right triangles, lattice point geometry, rational approximation, and dissection. Each chapter begins with the elementary math involved at its source, then discusses the development of the concepts and results in contemporary research. The book is meant for math students and teachers in high school through graduate school, and mathematicians. (Annotation ©2008 Book News Inc. Portland, OR)
Stochastic processes.
This is a brief introduction to stochastic processes for studying certain elementary continuous-time processes. After a description of the Poisson process and related processes with independent increments, as well as a brief look at Markov processes with a finite number of jumps, the book introduces Brownian motion and develops stochastic integrals and Itô's theory in the context of one-dimensional diffusion processes. The book ends with a brief survey of the general theory of Markov processes. Material is based on courses given by the author and can be used as a sequel to his book Probability Theory. Varadhan is affiliated with the Courant Institute of Mathematical Sciences at New York University. There is no subject index. (Annotation ©2008 Book News Inc. Portland, OR)
A (terse) introduction to linear algebra.
This sticks to the core of the study of vector spaces and the linear maps between them, starting with the idea of a finite-dimensional vector space to the canonical forms of linear operators and their matrices. Sections start with coverage of vector spaces, progressing to linear operators and matrices, the duality of vector spaces, determinants, invariant subspaces, inner-product spaces, structure theorems, and additional topics such as functions of an operator, quadratic forms, Perron-Frobenius theory, stochastic matrices, and representations of finite groups. The appendices include background material on equivalence relations-partitions, maps, groups, group actions, rings and algebras, and polynomials. This is intended for advanced undergraduates or others with an appropriate level of mathematical maturity. (Annotation ©2008 Book News Inc. Portland, OR)
Twenty-four hours of local cohomology.
Based on a June 2005 summer school, this series of 24 lectures introduces the algebraic sets, sheaf theory, and homological algebra leading to the definition and alternative characterizations of local cohomology. The graduate textbook illustrates how Cohen-Macaulay rings arise naturally, develops the Hartshorne-Lichtenbaum vanishing theorem, applies two classes of rings to polyhedral geometry, explains Grothendieck's duality theorem, and defines D-modules over rings of differential operators. (Annotation ©2008 Book News Inc. Portland, OR)
A view from the top; analysis, combinatorics, and number theory.
Iosevich (U. of Missouri) presents material from an upper division mathematics course he taught to synthesize techniques and ideas from the standard undergraduate mathematics curriculum. Among his topics are projections and cubes, a gentle entry into higher dimensions, and integer points and a crash course on Fourier analysis. Each chapter includes exercises and ends with notes, remarks, and difficult questions. No index is provided. (Annotation ©2008 Book News Inc. Portland, OR)
Yangians and classical Lie algebras.
Molev (University of Sydney) describes the structure and properties of Yangian and twisted Yangian associative algebras. The monograph proves classification theorems for the irreducible finite-dimensional representations of both algebras, develops explicit constructions of finite-dimensional irreducible representations of the Yangian, and applies Yangian theory to classical Lie algebras. Topics include the quantum determinant, the Sklyanin determinant, Gelfand-Tsetlin bases for representations, tensor products, Casimir elements, Capelli identities, and Mickelsson-Zhelobenko algebra. (Annotation ©2008 Book News Inc. Portland, OR)
