American Mathematical Society
Automorphic forms and applications.
Each year, the Institute for Advanced Study/Park City Mathematics Institute holds a graduate summer school. In 2002, the topic chosen for discussion was developments in analytic aspects of automorphic forms and their applications. This volume contains the lecture notes from the summer school and covers introductory issues, the basic theory of Eisenstein series, converse theorems and the Langlands- Shahidi method, Ramanujan conjectures and applications, analytic theory of GL(2) forms and L-functions, arithmetic quantum chaos, and unipotent flows on Gamma/G and applications. (Annotation ©2007 Book News Inc. Portland, OR)
Categories in algebra, geometry, and mathematical physics; proceedings.
These proceedings of the July 2005 conference and workshop include a significant number of papers commemorating the work of Ross Street. Topics include, rightly, the beginnings of category theory in Australia and higher gauge theory, the resolution of colored operads and rectification of homotropy algebras, simplical monoids and Segal categories, split extension classifiers and centrality, Moore normalization and Dold-Kan theorems for semi-abelian categories, branched coverings, degenerate categories and bi-categories, abstract cellularization, centers of monoidal categories of functions, ribbon categories, topological and conformal field theory as Frobenius algebras, characterization of torsion theories in general categories, weak units and homotopy three-types, quasi-categories and Segal spaces, parking functions, the K-theory, quantum field theories, three dimensional monad theory, orientals and nerves of complicated Gray-categories. Australia should be proud. (Annotation ©2007 Book News Inc. Portland, OR)
Collected papers of John Milnor; differential topology.
Milnor (mathematics, State U. of New York at Stony Brook) feels very fortunate to have started with the topology of manifolds in the 1950s; he was able to make very significant contributions to a fresh and dynamic field, and these papers prove it. Ranging from the later 1950s and into the later 1960s, these papers and include the "exotic spheres," including a procedure for killing homotopy groups of differentiable manifolds; expository lectures on topology, differentiable structures, and smooth manifolds with boundary based on "Variedades diferenciables con frontera", papers on relations with algebraic topology, and a series on cobiordism that is evidence of a staggering level of work done in a very short time. papers are in facsimile form and contain their original pagination. The author also provides a lively account of why he dedicated this volume to Georges De Rham, Hassler Whitney, and their belioved mountains. (Annotation ©2007 Book News Inc. Portland, OR)
Combinatorial problems and exercises, 2d ed.
Lovász provides extensive help to those wishing to learn existing techniques in combinatories, approaching the topic in a participatory lecture format and providing hundreds of progressive exercises. In fact, his approach is to have student learn almost entirely from the exercises, which he has updated extensively from the previous edition. Along with the exercises come both hints and solutions as he works through basic enumeration, the sieve process, permutations, classical enumeration problems in graph theory, parity and duality, connectivity, factors of graphs, independent sets of points, chromatic number, problems for graphs, the spectra of graphs and random walks, automorphisms of graphs, hypergraphs, Ramsey theory and reconstruction. Students beginning in graph theory, combinatories or their applications will find this particularly useful, as will researchers who wish to apply the concepts to their studies in mathematics, computer science, management science, or electrical engineering. (Annotation ©2007 Book News Inc. Portland, OR)
Cones and duality.
In the last 100 years or so researchers have worked with ordered vector spaces and cones, often alongside their work in the different perspective of functional analysis and operator theory. About 50 years ago mathematicians from a range of schools worldwide started studying ordered vector spaces systematically, often to solve optimization problems. As a result this field has become integral to engineering, economics and the social sciences. Here the authors consider cones anew and show how their ideas apply to various fields, aiding researchers and graduate students in mathematics, economics and finance. They cover cones, cones in topological vector spaces, Yudin and pull-back cones, Krein operators, K-lattices, order extensions, piecewise affine functions and in a fascinating appendix, linear topologies. (Annotation ©2007 Book News Inc. Portland, OR)
Elements of the homology theory.
Prasolov developed from his seminars on topology for second-year graduate students at the Independent U. of Moscow, so his treatment of homology and cohomology takes a very structured form rather than emerging as a series of mathematical vignettes. Prasolov starts with the definition of simplicial homology and cohomology and backs this up with examples and applications, describes calculations, the Euler characteristic and the Lefschetz theorem. He then introduces cohomology rings in terms of the Kolmogotov-Alexander multiplication in cohomology, the homology and cohomology of manifolds, and the Künneth theorem, then turns to applications of simplicial homology, including homology's relationship with homotopy, characteristic classes, group actions and Stenrod squares, singular homology, Cech cohomology and de Rham cohomology. Other topics include the Alexander polynomial, the Arf invariant, embeddings and immersions, complex manifolds, Lie groups and H-spaces. Prasolov includes solutions for selected exercises. (Annotation ©2007 Book News Inc. Portland, OR)
Ergodic theory and related fields; proceedings.
This volume collects seven contributions from participants of the February 2004, 2005, and 2006 Chapel Hill ergodic theory workshops. The mathematicians investigate the pointwise convergence of weighted averages linked to averages along cubes, divergent ergodic averages along the squares, the one-sided ergodic Hilbert transform, deterministic walks in Markov environments with constant rigidity, limit theorems for sequential expanding dynamical systems, and random Fourier-Stieltjes transforms. No index is provided. (Annotation ©2007 Book News Inc. Portland, OR)
Foliations in Cauchy-Riemann geometry.
In this study of the relationship between foliation theory and differential geometry and analysis on Cauchy-Riemann (CR) manifolds, the main objects are transversally and tangentially CR foliations, Levi foliations of CR manifolds, solutions of the Yang-Mills equations, tangentially Monge-Ampere foliations, the transverse Beltrami equations and CR manifolds. This is a novel, multidisciplinary approach that uses the methods of foliation theory for specific applications. Punctuated by open problems designed to pique the interest of mathematicians, this includes such topics as foliated Lorentz manifolds, holomorphic extensions of Levi foliations, analysis on pseudoconvex domains, CR submanifolds of maximal CR dimension, the Graham-Lee connection, flows, degenerate CR manifolds, and a review of orbifold theory. (Annotation ©2007 Book News Inc. Portland, OR)
Functions of several complex variables and their singularities.
Ebeling introduces the theory of functions of several complex variables and their singularities, emphasizing the topological aspects. He provides all the necessary prerequisites for graduate students and practitioners, describing Riemann surfaces (including coverings, analytical continuation, and Puiseaux expansion), holomorphic functions of several variables (including analytic sets and analytic set germs as well as regular and singular points of analytic sets), isolated singularities of holomorphic functions (including isolated critical points and the universal unfolding), fundamentals of differential topology (including singular homology groups and linking numbers), and the topology of singularities (including the Picard-Lefschetz theorem, the Milnor fibration, the Coxeter-Dynkin diagram, the Selfert form and the action of the braid group. (Annotation ©2007 Book News Inc. Portland, OR)
Geometric analysis on the Heisenberg group and its generalizations.
Although designed as a course or seminar text for graduate students interested in developments in the subReimannian manifolds (manifolds with the Heisenberg principle built in) and sub-elliptic operators theory, this also works as a resource for pure and applied mathematicians and theoretical physics working in quantum mechanics. One of the authors' most interesting innovations is introducing the complex Hamiltonian mechanics techniques and use them to describe the fundamental solutions and heat propagators in quantum mechanics. They introduce geometric mechanics on the Heisenberg group, then give geometric analyses of the step 4 case, the step 2(k+1) case, the geometry of higher dimensional Heisenberg groups, complex Hamiltonian mechanics and quantum mechanics on the Hiesenberg group. The result is fresh and lively while also being thorough. The authors provide exercises for each chapter. (Annotation ©2007 Book News Inc. Portland, OR)
Geometric and topological methods for quantum field theory; proceedings.
Based upon lectures and other communications at the July 2005 summer school, this introduces readers to some recent developments in active research on the interface between geometry, topology and quantum field theory. In five survey lectures the contributors cover anomalies and noncommutative geometry, deformation quantization and Poisson algebras, and topological quantum field theory and orbifolds. These are followed by nine cutting-edge articles with topics including n-flat connections, Dirac equations in a black hole background, homological matrices, quantitative properties of stratified flows, property (T) and tensor products by certain irreducible finite dimensional representations, Painleve equations for invariant instantons, quantum statistical mechanics and class field theory, Kashiwara's quantization of complex contact manifolds and K-theoretic labeling for quasicrystals. (Annotation ©2007 Book News Inc. Portland, OR)
Golden years of Moscow mathematics, 2d ed.
Mathematicians who were based in Moscow remember the profession as it was practiced and taught during the Soviet era. Their perspectives include the Moscow school of the theory of functions during the 1930s, A. N. Kolmogorov (b. 1903), life and automorphic forms in the Soviet Union, and Soviet mathematics of the 1950s and 1960s. A new essay summarizes developments since the early 1990s when the first edition appeared. Only names are indexed. (Annotation ©2007 Book News Inc. Portland, OR)
High-dimensional partial differential equations in science and engineering; proceedings.
Until the genesis of computational resources sufficient to handle them, these equations have been a stumbling block in many fields of study. Now, however, they can be managed by combining recent developments in numerical and computational techniques and the use of computers with parallel structures. Taken from a meeting held at the U. of Montreal in August 2005, these papers describe the many new applications these advances have made, including kinetic plasma physics equations, the many-body Schrödinger equation, Dirac and Maxwell equations for molecular electronic structure and nuclear dynamic computations, options pricing equations in mathematical finance and Fokker-Planck and fluid dynamics equations for complex fluids. (Annotation ©2007 Book News Inc. Portland, OR)
In the tradition of Ahlfors-Bers; proceedings.
Lars Ahlfors and Lipman Bers created significant mathematical legacies within the fields of algebraic geometry, mathematical physics, dynamics, geometric group theory, number theory and topology. In these proceedings of the May 2005 colloquium that bears their names, contributors present new research and also some expository material in such studies as uniformly exponential growth and mapping of class groups of surfaces, curvature and stretchiness and their relation to dynamics, some special loci in the Segal space of genus two, a new approach to the automorphism theorem for Teichmüller spaces, the energy of twisted harmonic maps of Riemann spaces, small eigenvalues and maximal laminations on complete surfaces of negative curvature, a generalized hyperbolic metric for plane domains, random hyperbolic surfaces and measured laminations and, as befitting, a survey of the complete analyst's traveling salesman theorems. (Annotation ©2007 Book News Inc. Portland, OR)
Operator theory in function spaces, 2d ed.
Zhu (mathematics. State U. of New York at Albany) covers Toeplitz operators, Hankel operators and composition operators on both Bergman space and hardy space. His setting is the unit disk and he emphasizes size estimates of these operators in terms of boundedness, compactness, and membership in the Schatten classes. His results concern the relationship between operator-theoretic properties of these operators and function-theoretic properties of the inducing symbols. Writing for research mathematicians and graduate students in complex analysis and operator theory, Zhue begins with bounded linear operators, continuing with the interpolation of Banach spaces, integral operators on Lp spaces, Bergman spaces, Bloch and Besov spaces, the Berezin transform, Toeplitz operators on the Bergman space, Hangel operators on the Bergman space, Hardy spaces and BMO, Hankel operators on the Hardy space, and the aforesaid composition operators. Zhue provides exercises with each section. (Annotation ©2007 Book News Inc. Portland, OR)
Probability and mathematical physics; a volume in honor of Stanislav Molchanov.
Molchanov (mathematics, North Carolina U.) accomplished much (he has published over 250 papers), and not the least of which was his participation in the development of the intermittency theory for non-stationary random particles. These papers by friends, students and colleagues indicate the range of his study and his generosity, as the cover such topics as some new estimates on the spectral shift functions associated with random Schrödinger operations, asymptotics of the Poincaré functions, localization of low energies for attractive Poisson random Schrödinger operators, Fermi-Dirac generators and tests for randomness, a hierarchical Anderson model, Green's functions of generalized Laplacians and orthogonal polynomials with exponentially decaying recursion coefficients. The editors include a biography of Molchanov. (Annotation ©2007 Book News Inc. Portland, OR)
Quantum groups; proceedings.
Donin was published before he completed his MA at Moscow State U. and again before he completed his Ph.D. He proposed a new approach based on the Banach manifold and the Banach Lie group techniques of Douady. His later work on deformations of homomorphic bundles over complex spaces with singularities and versal deformations of germs of complex spaces was even more distinguished. These proceedings of the July 2004 conference reflect the high regard in which contributors held Donin, and include a survey of his research along with such topics as bicrystals and crystal bases, the small quantum group and the Springer resolution, Fourier transforms for Hopf algebras, quantization, basic representations of quantum current algebras in higher genus, Poincare-Birkhoff-Witt expansions of the canonical elliptic differential form, the Drinfeld double for orbifolds, symmetrically factorizable groups and set-theoretical solutions for the pentagon equation, the dynamical reflection equation and Carter-Rieger-Saito movies. (Annotation ©2007 Book News Inc. Portland, OR)
Recurrence and topology.
Alongi (mathematics and computer science, Northwestern University) and Nelson (mathematics, Carleton College) seek, along with Poisson and generations after him, what it means for a solution of a differential equation to be recurrent. Their approach develops increasingly more general topological modes of recurrence for dynamic situations beginning with fixed points and concluding with chain recurrent points. They include extensive examples as they cover flows, recurrent points, irreducible sets and test functions. They also include appendices on discrete dynamical systems, circle rotations and the Hausdorff metric and a range of exercises for each chapter. (Annotation ©2007 Book News Inc. Portland, OR)
Recent trends in coding theory and its applications.
Coding theory includes a wide range of mathematical topics, both pure and applied, a fact which is reflected in the contributions here, which include algebraic geometry codes such as Elkes's modularity conjecture and explicit towers and codes, improved algebraic geometry bonds and upper and lower bounds for A(q), graph-based codes such as Reed Muller codes and symplectic geometry, and quantum codes including a new description of quantum error-correcting code. (Annotation ©2007 Book News Inc. Portland, OR)
Stochastic analysis and partial differential equations; emphasis year 2004-2005 on stochastic analysis and partial differential equations, Northwestern University, Evanston, Illinois.
This volume consists of 16 original research papers and expository articles written by speakers invited to the mathematics seminars and June 2005 conference at Northwestern University. The researchers report recent progress in the stochastic analysis of turbulent mixing, construct weak solutions of the Navier-Stokes equations, and investigate the spectral properties of subordinate processes in domains. Other topics include homogenization of stochastic Hamilton-Jacobi equations, general relative entropy in a nonlinear McKendrick model, pointwise Fourier inversion in analysis and geometry, and a class of one-dimensional Markov processes with continuous paths. (Annotation ©2007 Book News Inc. Portland, OR)
