Böcker i London Mathematical Society Lecture Note Series-serien

This is the first book to deal with invariant theory and the representations of finite groups. By restricting attention to finite groups Dr Benson is able to avoid recourse to the technical machinery of algebraic groups, and he develops the necessary results from commutative algebra as he proceeds. Thus the book should be accessible to graduate students. In detail, the book contains an account of invariant theory for the action of a finite group on the ring of polynomial functions on a linear representation, both in characteristic zero and characteristic p. Special attention is paid to the role played by pseudoreflections, which arise because they correspond to the divisors in the polynomial ring which ramify over the invariants. Also included is a new proof by Crawley-Boevey and the author of the Carlisle-Kropholler conjecture. This volume will appeal to all algebraists, but especially those working in representation theory, group theory, and commutative or homological algebra.

Block theory is a part of the theory of modular representation of finite groups and deals with the algebraic structure of blocks. In this volume Burkhard Kulshammer starts with the classical structure theory of finite dimensional algebras, and leads up to Puigs main result on the structure of the so called nilpotent blocks, which he discusses in the final chapter. All the proofs in the text are given clearly and in full detail, and suggestions for further reading are also included. For researchers and graduate students interested in group theory or representation theory, this book will form an excellent self contained introduction to the theory of blocks.

This is an introduction to, and survey of, the constructive approaches to pure mathematics. The authors emphasise the viewpoint of Errett Bishop's school, but intuitionism. Russian constructivism and recursive analysis are also treated, with comparisons between the various approaches included where appropriate. Constructive mathematics is now enjoying a revival, with interest from not only logicans but also category theorists, recursive function theorists and theoretical computer scientists. This account for non-specialists in these and other disciplines.

The study of sets of uniqueness for trigonometric series has a long history, originating in the work of Riemann, Heine, and Cantor in the mid-nineteenth century. Since then it has been a fertile ground for numerous investigations involving real analysis, classical and abstract harmonic analysis, measure theory, functional analysis and number theory. In this book are developed the intriguing and surprising connections that the subject has with descriptive set theory. These have only been discovered recently and the authors present here this novel theory which leads to many new results concerning the structure of sets of uniqueness and include solutions to some of the classical problems in this area. In order to make the material accessible to logicians, set theorists and analysts, the authors have covered in some detail large parts of the classical and modern theory of sets of uniqueness as well as the relevant parts of descriptive set theory. Thus the book is essentially self-contained and will make an excellent introduction to the subject for graduate students and research workers in set theory and analysis.

This book presents an account of several conjectures arising in commutative algebra from the pioneering work of Serre and Auslander-Buchsbaum. The approach is via Hochster's 'Big Cohen-Macaulay modules', though the complementary view point of Peskine-Szpiro and Roberts, who study the homology of certain complexes, is not neglected. Various refinements of Hochster's construction, obtained in collaboration with Bartijn, are included. A special feature is a long chapter written by Van den Dries which explains how a certain type of result can be 'lifted' from prime characteristic to characteristic zero. Though this is primarily a research monograph, it does provide introductions to most of the topics treated. Non-experts may therefore find it an appealing guide into an active area of algebra.

In this volume the authors seek to illustrate how methods of differential geometry find application in the study of the topology of differential manifolds. Prerequisites are few since the authors take pains to set out the theory of differential forms and the algebra required. The reader is introduced to De Rham cohomology, and explicit and detailed calculations are present as examples. Topics covered include Mayer-Vietoris exact sequences, relative cohomology, Pioncare duality and Lefschetz's theorem. This book will be suitable for graduate students taking courses in algebraic topology and in differential topology. Mathematicians studying relativity and mathematical physics will find this an invaluable introduction to the techniques of differential geometry.

This self-contained and relatively elementary introduction to functions of several complex variables and complex (especially compact) manifolds is intended to be a synthesis of those topics and a broad introduction to the field. Part I is suitable for advanced undergraduates and beginning postgraduates whilst Part II is written more for the graduate student. The work as a whole will be useful to professional mathematicians or mathematical physicists who wish to acquire a working knowledge of this area of mathematics. Many exercises have been included and indeed they form an integral part of the text. The prerequisites for understanding Part I would be met by any mathematics student with a first degree and together the two parts provide an introduction to the more advanced works in the subject.

The core model, K, is a generalization of Godel's constructible universe of set theory; K is used to produce 'fine structural' results of a less restrictive kind. This book aims to introduce the core model to those with a basic knowledge of axiomatic set theory. The covering lemma for K is the main technical result but other applications are also considered. The author gives a full exposition of general fine structure and of iterated ultrapowers and concludes the work with a short section on the difficulties encountered in constructing more general core models using 'extenders'.

The book is written in three parts. Part I consists of preparatory work on algebras, needed in Parts II and III. This material is presented in a classical, though unusual, way. Part II consists of a modern description of the theory of Brauer groups over fields (from as elementary a point of view as possible). Part III covers some new developments in the theory which, until now, have not been available except in journals. The principal topic discussed in this section is reduced K,-theory. This book will be of interest to graduate students in pure mathematics and to professional mathematicians.

As a result of the work of the nineteenth-century mathematician Arthur Cayley, algebraists and geometers have extensively studied permutation of sets. In the special case that the underlying set is linearly ordered, there is a natural subgroup to study, namely the set of permutations that preserves that order. In some senses. these are universal for automorphisms of models of theories. The purpose of this book is to make a thorough, comprehensive examination of these groups of permutations. After providing the initial background Professor Glass develops the general structure theory, emphasizing throughout the geometric and intuitive aspects of the subject. He includes many applications to infinite simple groups, ordered permutation groups and lattice-ordered groups. The streamlined approach will enable the beginning graduate student to reach the frontiers of the subject smoothly and quickly. Indeed much of the material included has never been available in book form before, so this account should also be useful as a reference work for professionals.

Line graphs have the property that their least eigenvalue is greater than or equal to -2, a property shared by generalized line graphs and a finite number of so-called exceptional graphs. This book deals with all these families of graphs in the context of their spectral properties. The authors discuss the three principal techniques that have been employed, namely 'forbidden subgraphs', 'root systems' and 'star complements'. They bring together the major results in the area, including the recent construction of all the maximal exceptional graphs. Technical descriptions of these graphs are included in the appendices, while the bibliography provides over 250 references. This will be an important resource for all researchers with an interest in algebraic graph theory.

A thorough account of the methods that underlie the theory of subalgebras of finite von Neumann algebras, this book contains a substantial amount of current research material and is ideal for those studying operator algebras. The conditional expectation, basic construction and perturbations within a finite von Neumann algebra with a fixed faithful normal trace are discussed in detail. The general theory of maximal abelian self-adjoint subalgebras (masas) of separable II1 factors is presented with illustrative examples derived from group von Neumann algebras. The theory of singular masas and Sorin Popa's methods of constructing singular and semi-regular masas in general separable II1 factor are explored. Appendices cover the ultrapower of a II1 factor and the properties of unbounded operators required for perturbation results. Proofs are given in considerable detail and standard basic examples are provided, making the book understandable to postgraduates with basic knowledge of von Neumann algebra theory.

The arithmetic properties of modular forms and elliptic curves lie at the heart of modern number theory. This book develops a generalisation of the method of Euler systems to a two-variable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and Swinnerton-Dyer (BSD) formula. Three main steps are outlined: the first is to parametrise 'big' cohomology groups using (deformations of) modular symbols. Finiteness results for big Selmer groups are then established. Finally, at weight two, the arithmetic invariants of these Selmer groups allow the control of data from the BSD conjecture. As the first book on the subject, the material is introduced from scratch; both graduate students and professional number theorists will find this an ideal introduction. Material at the very forefront of current research is included, and numerical examples encourage the reader to interpret abstract theorems in concrete cases.

The Navier-Stokes equations were firmly established in the 19th Century as the system of nonlinear partial differential equations which describe the motion of most commonly occurring fluids in air and water, and since that time exact solutions have been sought by scientists. Collectively these solutions allow a clear insight into the behavior of fluids, providing a vehicle for novel mathematical methods and a useful check for computations in fluid dynamics, a field in which theoretical research is now dominated by computational methods. This 2006 book draws together exact solutions from widely differing sources and presents them in a coherent manner, in part by classifying solutions via their temporal and geometric constraints. It will prove to be a valuable resource to all who have an interest in the subject of fluid mechanics, and in particular to those who are learning or teaching the subject at the senior undergraduate and graduate levels.

Finite groups of Lie type encompass most of the finite simple groups. Their representations and characters have been studied intensively for half a century, though some key problems remain unsolved. This is the first comprehensive treatment of the representation theory of finite groups of Lie type over a field of the defining prime characteristic. As a subtheme, the relationship between ordinary and modular representations is explored, in the context of Deligne-Lusztig characters. One goal has been to make the subject more accessible to those working in neighbouring parts of group theory, number theory, and topology. Core material is treated in detail, but the later chapters emphasize informal exposition accompanied by examples and precise references.

This is an introduction to non-commutative geometry, with special emphasis on those cases where the structure algebra, which defines the geometry, is an algebra of matrices over the complex numbers. Applications to elementary particle physics are also discussed. This second edition is thoroughly revised and includes new material on reality conditions and linear connections plus examples from Jordanian deformations and quantum Euclidean spaces. Only some familiarity with ordinary differential geometry and the theory of fibre bundles is assumed, making this book accessible to graduate students and newcomers to this field.

Semiclassical approximation addresses the important relationship between quantum and classical mechanics. There has been a very strong development in the mathematical theory, mainly thanks to methods of microlocal analysis. This book develops the basic methods, including the WKB-method, stationary phase and h-pseudodifferential operators. The applications include results on the tunnel effect, the asymptotics of eigenvalues in relation to classical trajectories and normal forms, plus slow perturbations of periodic Schrodinger operators appearing in solid state physics. No previous specialized knowledge in quantum mechanics or microlocal analysis is assumed, and only general facts about spectral theory in Hilbert space, distributions, Fourier transforms and some differential geometry belong to the prerequisites. This book is addressed to researchers and graduate students in mathematical analysis, as well as physicists who are interested in rigorous results. A fairly large fraction can be (and has been) covered in a one semester course.

This book provides a detailed but concise account of the theory of structure of finite p-groups admitting p-automorphisms with few fixed points. The relevant preliminary material on Lie rings is introduced and the main theorems of the book on the solubility of finite p-groups are then presented. The proofs involve notions such as viewing automorphisms as linear transformations, associated Lie rings, powerful p-groups, and the correspondences of A. I. Mal'cev and M. Lazard given by the Baker-Hausdorff formula. Many exercises are included. This book is suitable for graduate students and researchers working in the fields of group theory and Lie rings.

The James space J and the James tree space JT were constructed as counterexamples to several outstanding conjectures in Banach space theory. This book is a compendium of most of the known results about these spaces, frequently taken from the original sources, but presented in a unified and up to date fashion. Generalisations of J and JT are also discussed and other pathological Banach spaces are introduced. Specialists will welcome this book for its drawing together of classical material and recent results. Graduate students should also find that this book offers an excellent introduction to more advanced topics in Banach space theory.

Subfactors have been a subject of considerable research activity for about fifteen years and are known to have significant relations with other fields such as low dimensional topology and algebraic quantum field theory. These notes give an introduction to the subject suitable for a student who has only a little familiarity with the theory of Hilbert space. A new pictorial approach to subfactors is presented in a late chapter.

In this book the authors present their research into the foundations of the theory of Polish groups and the associated orbit equivalence relations. The particular case of locally compact groups has long been studied in many areas of mathematics. Non-locally compact Polish groups occur naturally as groups of symmetries in such areas as logic (especially model theory), ergodic theory, group representations, and operator algebras. Some of the topics covered here are: topological realizations of Borel measurable actions; universal actions; applications to invariant measures; actions of the infinite symmetric group in connection with model theory (logic actions); dichotomies for orbit spaces (including Silver, Glimm-Effros type dichotomies and the topological Vaught conjecture); descriptive complexity of orbit equivalence relations; definable cardinality of orbit spaces.

This book is devoted to the asymptotic properties of solutions of stochastic evolution equations in infinite dimensional spaces. It is divided into three parts: Markovian dynamical systems; invariant measures for stochastic evolution equations; invariant measures for specific models. The focus is on models of dynamical processes affected by white noise, which are described by partial differential equations such as the reaction-diffusion equations or Navier-Stokes equations. Besides existence and uniqueness questions, special attention is paid to the asymptotic behaviour of the solutions, to invariant measures and ergodicity. Some of the results found here are presented for the first time. For all whose research interests involve stochastic modelling, dynamical systems, or ergodic theory, this book will be an essential purchase.

There is now a large body of theory concerning algebraic varieties over finite fields, and many conjectures exist in this area that are of great interest to researchers in number theory and algebraic geometry. This book is concerned with the arithmetic of diagonal hypersurfaces over finite fields, with special focus on the Tate conjecture and the Lichtenbaum-Milne formula for the central value of the L-function. It combines theoretical and numerical work, and includes tables of Picard numbers. Although this book is aimed at experts, the authors have included some background material to help non-specialists gain access to the results.

Hilbert C*-modules are objects like Hilbert spaces, except that the inner product, instead of being complex valued, takes its values in a C*-algebra. The theory of these modules, together with their bounded and unbounded operators, is not only rich and attractive in its own right but forms an infrastructure for some of the most important research topics in operator algebras. This book is based on a series of lectures given by Professor Lance at a summer school at the University of Trondheim. It provides, for the first time, a clear and unified exposition of the main techniques and results in this area, including a substantial amount of new and unpublished material. It will be welcomed as an excellent resource for all graduate students and researchers working in operator algebras.

This is an introductory text intended to give the non-specialist a comprehensive insight into the science of biotransformations. The book traces the history of biotransformations, clearly spells out the pros and cons of conducting enzyme-mediated versus whole-cell bioconversions, and gives a variety of examples wherein the bio-reaction is a key element in a reaction sequence leading from cheap starting materials to valuable end products.

This two-volume book contains selected papers from the international conference 'Groups 1993 Galway/St Andrews' which was held at University College, Galway in August 1993. The wealth and diversity of group theory is represented in these two volumes. Five main lecture courses were given at the conference. These were 'Geometry, Steinberg representations and complexity' by J. L. Alperin (Chicago), 'Rickard equivalences and block theory' by M. Broue (ENS, Paris), 'Cohomological finiteness conditions', by P. H. Kropholler (Queen Mary and Westfield College, London), 'Counting finite index subgroups', by A. Lubotzky (Hebrew University, Jerusalem), 'Lie methods in group theory' by E. I. Zel'manov (University of Wisconsin at Madison). Articles based on their lectures, in one case co-authored, form a substantial part of the Proceedings. Another main feature of the conference was a GAP workshop jointly run by J. Neubuser and M. Schonert (Rheinisch-Westfalische Technische Hochschole, Aachen). Two articles by Professor Neubuser, one co-authored, appear in the Proceedings. The other articles in the two volumes comprise both refereed survey and research articles contributed by other conference participants. As with the Proceedings of the earlier 'Groups-St Andrews' conferences it is hoped that the articles in these Proceedings will, with their many references, prove valuable both to experienced researchers and also to new postgraduates interested in group theory.

This two-volume book contains selected papers from the international conference 'Groups 1993 Galway/St Andrews' which was held at University College Galway in August 1993. The wealth and diversity of group theory is represented in these two volumes. Five main lecture courses were given at the conference. These were 'Geometry, Steinberg representations and complexity' by J. L. Alperin (Chicago), 'Rickard equivalences and block theory' by M. Broue (ENS, Paris), 'Cohomological finiteness conditions' by P. H. Kropholler (QMW, London), 'Counting finite index subgroups' by A. Lubotzky (Hebrew University, Jerusalem), 'Lie methods in group theory' by E. I. Zel'manov (University of Wisconsin at Madison). Articles based on their lectures, in one case co-authored, form a substantial part of the Proceedings. Another main feature of the conference was a GAP workshop jointly run by J. Neubuser and M. Schonert (RWTH, Aachen). Two articles by Professor Neubuser, one co-authored, appear in the Proceedings. The other articles in the two volumes comprise both refereed survey and research articles contributed by other conference participants. As with the Proceedings of the earlier 'Groups-St Andrews' conferences it is hoped that the articles in these Proceedings will, with their many references, prove valuable both to experienced researchers and also to new postgraduates interested in group theory.

This book describes work, largely that of the author, on the characterization of closed 4-manifolds in terms of familiar invariants such as Euler characteristic, fundamental group, and Stiefel-Whitney classes. Using techniques from homological group theory, the theory of 3-manifolds and topological surgery, infrasolvmanifolds are characterized up to homeomorphism, and surface bundles are characterized up to simple homotopy equivalence. Non-orientable cases are also considered wherever possible, and in the final chapter the results obtained earlier are applied to 2-knots and complex analytic surfaces. This book is essential reading for anyone interested in low-dimensional topology.

The subject of this book lies on the boundary between probability theory and the theory of function spaces. Here Professor Braverman investigates independent random variables in rearrangement invariant (r.i.) spaces. The significant feature of r.i. spaces is that the norm of an element depends on its distribution only, and this property allows the results and methods associated with r.i. spaces to be applied to problems in probability theory. On the other hand, probabilistic methods can also prove useful in the study of r.i. spaces. In this book new techniques are used and a number of interesting results are given. Most of the results are due to the author but have never before been available in English. Here they are all presented together in a volume that will be essential reading for all serious researchers in this area.

This is the first unified treatment in book form of the lower K-groups of Bass and the lower L-groups of the author. These groups arise as the Grothendieck groups of modules and quadratic forms which are components of the K- and L-groups of polynomial extensions. They are important in the topology of non-compact manifolds such as Euclidean spaces, being the value groups for Whitehead torsion, the Siebemann end obstruction and the Wall finiteness and surgery obstructions. Some of the applications to topology are included, such as the obstruction theories for splitting homotopy equivalences and for fibering compact manifolds over the circle. Only elementary algebraic constructions are used, which are always motivated by topology. The material is accessible to a wide mathematical audience, especially graduate students and research workers in topology and algebra.