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The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "e;CAT(0) groups."e; The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincare Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.

By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary. This book explains in detail the following topics about PEL-type Shimura varieties and their compactifications: A construction of smooth integral models of PEL-type Shimura varieties by defining and representing moduli problems of abelian schemes with PEL structures An analysis of the degeneration of abelian varieties with PEL structures into semiabelian schemes, over noetherian normal complete adic base rings A construction of toroidal and minimal compactifications of smooth integral models of PEL-type Shimura varieties, with detailed descriptions of their structure near the boundary Through these topics, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai).

This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, Zhen-Qing Chen and Masatoshi Fukushima cover the essential elements and applications of the theory of symmetric Markov processes, including recurrence/transience criteria, probabilistic potential theory, additive functional theory, and time change theory. The authors develop the theory in a general framework of symmetric quasi-regular Dirichlet forms in a unified manner with that of regular Dirichlet forms, emphasizing the role of extended Dirichlet spaces and the rich interplay between the probabilistic and analytic aspects of the theory. Chen and Fukushima then address the latest advances in the theory, presented here for the first time in any book. Topics include the characterization of time-changed Markov processes in terms of Douglas integrals and a systematic account of reflected Dirichlet spaces, and the important roles such advances play in the boundary theory of symmetric Markov processes. This volume is an ideal resource for researchers and practitioners, and can also serve as a textbook for advanced graduate students. It includes examples, appendixes, and exercises with solutions.

This book seeks to describe the rapid development in recent decades of sieve methods able to detect prime numbers. The subject began with Eratosthenes in antiquity, took on new shape with Legendre's form of the sieve, was substantially reworked by Ivan M. Vinogradov and Yuri V. Linnik, but came into its own with Robert C. Vaughan and important contributions from others, notably Roger Heath-Brown and Henryk Iwaniec. Prime-Detecting Sieves breaks new ground by bringing together several different types of problems that have been tackled with modern sieve methods and by discussing the ideas common to each, in particular the use of Type I and Type II information. No other book has undertaken such a systematic treatment of prime-detecting sieves. Among the many topics Glyn Harman covers are primes in short intervals, the greatest prime factor of the sequence of shifted primes, Goldbach numbers in short intervals, the distribution of Gaussian primes, and the recent work of John Friedlander and Iwaniec on primes that are a sum of a square and a fourth power, and Heath-Brown's work on primes represented as a cube plus twice a cube. This book contains much that is accessible to beginning graduate students, yet also provides insights that will benefit established researchers.

This is the first comprehensive reference published on heat equations associated with non self-adjoint uniformly elliptic operators. The author provides introductory materials for those unfamiliar with the underlying mathematics and background needed to understand the properties of heat equations. He then treats Lp properties of solutions to a wide class of heat equations that have been developed over the last fifteen years. These primarily concern the interplay of heat equations in functional analysis, spectral theory and mathematical physics. This book addresses new developments and applications of Gaussian upper bounds to spectral theory. In particular, it shows how such bounds can be used in order to prove Lp estimates for heat, Schrodinger, and wave type equations. A significant part of the results have been proved during the last decade. The book will appeal to researchers in applied mathematics and functional analysis, and to graduate students who require an introductory text to sesquilinear form techniques, semigroups generated by second order elliptic operators in divergence form, heat kernel bounds, and their applications. It will also be of value to mathematical physicists. The author supplies readers with several references for the few standard results that are stated without proofs.

Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years. Log-Gases and Random Matrices gives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials. Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variable; the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painleve transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory. This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, making Log-Gases and Random Matrices an indispensable reference work, as well as a learning resource for all students and researchers in the field.

The aim of the book is to give necessary and sufficient conditions for a map to be stable. This is achieved in a wide range of dimensions via a detailed study of the geometry and topology of many classes of "generic" singularities.

The central part of the pluripotential theory is occupied by maximal plurisubharmonic functions and the generalized complex Monge-Ampere operator. The interplay between these two notions provides the focal point of this monograph, which explores classical properties of subharmonic and plurisubharmonic functions.

Although Lie polynomials first appeared at the turn of the century, there have been many recent developments especially from the point of view of representation theory. This book covers all aspects, with emphasis on the algebraic and combinatorial point of view as well as representation theory.

The book describes many specific classes of Banach algebras, including function algebras, group algebras, algebras of operators, C*=algebras, and radical Banach algebras; it is a compendium of results on these examples. The subject interweaves algebras, functional analysis, and complex analysis, and has a dash of set theory and logic; the background in all these areas is fully explained.

This work summarises the development of a classification system of finite p-groups. The authors provide a careful summary and explanation of the many and difficult original research papers on the co-class conjecture and the structure theorem, thus elucidating the background research for those new to the area as well as for experienced researchers.

Finite Coxeter groups and related structures arise naturally in several branches of mathematics, for example, Lie algebras or theory of knots and links. This is the first book which develops the character theory of finite Coxeter groups and Iwahori-Hecke algebras in a systematic way, ranging from classical results to recent developments.

Profinite groups are of interest to mathematicians in a variety of areas, including number theory, abstract groups, and analysis. This text provides an introduction to the subject and is designed to convey basic facts and enable readers to enhance their skills in manipulating profinite groups.

This book is a modern treatment of a classical area of operator theory. Written in a meticulous and detailed style, with the modern graduate student of analysis in mind, it contains many simplifications of existing literature. It is full of new results, as well as many illuminating examples. Carefully cross referenced throughout, it also includes an extensive list of the relevant literature.

A revised and expanded second edition of Reiter's Classical Harmonic Analysis and Locally Compact Groups (Clarendon Press 1968).

This work contains the first explicit construction of hidden structures (mantles and trains). The author shows how many infinite-dimensional groups are in fact only a small part of a much larger object, analogous to the way real numbers are embedded within complex numbers.

This monograph uses the language of homological algebra and sheaf theory to describe both classical results and recent developments in the spectral theory of linear operators. It draws together concepts from function theory and complex analytical geometry.

This advanced text expounds the established theory of ordered fields, and continues to develop a quite original theory of super-real fields. This theory has important applications in analysis and logic.

This invaluable reference tool is the first to present the general theory of algebras of operators on a Hilbert space, and the modules over such algebras. The new theory of operator spaces is presented early on and the text assembles the basic concepts, theory and methodologies needed to equip a beginning researcher in this area.

Highly topical and original monograph, introducing the author's work on the Riemann zeta function and its adelic interpretation of interest to a wide range of mathematicians and physicists.

This volume is concerned with the application of exponential sum techniques to a variety of problems in number theory, in particular the Riemann Zeta Function and the problem of estimating the number of lattice points in regions.

This book deals with the arithmetical properties of almost all real numbers. It brings together many different types of result (including normal numbers, Diophantine approximation and uniform distribution) never covered within the same volume before. By this approach interactions and common themes between different branches of the subject are revealed.

This is a reissue of a classic text previously published by the LMS, aimed at beginning postgraduate students in algebra and number theory. It gives a well-paced introduction to topics central to several active areas of mathematical research, and provides a very helpful background reference to researchers.

This book explores the application of mathematical analysis to problems of interpolation and engineering, including systems identification, and signal processing and sampling. It develops the mathematical background, and includes theoretical results and practical material on input design and identification algorithms. Many topics are presented for the first time.

Presents easy to understand proofs of some of the most difficult results about polynomials demonstrated by means of applications.

A comprehensive account of the restricted Burnside problem, including a new chapter on the highly acclaimed and recent work from E.I. Zelmanov.

This book explores in detail the connections between self-duality and integrability, and also the application of twistor techniques to integrable systems.

This concise introduction to the subject emphasizes the various classes of regular semigroups. More than 150 exercises, accompanied by references to the relevant research literature, direct readers to areas not explicitly covered in the text.

An introduction to the theory of existentially closed groups, this study is presented from a group theoretical, rather than a model theoretical, point of view.