Bücher der Reihe London Mathematical Society Lecture Note Series

This book provides an introduction to the field of linear algebraic monoids. This subject represents a synthesis of ideas from the theory of algebraic groups, algebraic geometry, matrix theory and abstract semigroup theory. Since every representation of an algebraic group gives rise to an algebraic monoid, the objects of study do indeed arise naturally.

Throughout the academic year 1986-87, the University of Illinois was host to a symposium on mathematical analysis which was attended by some of the leading figures in the field. This book arises out of this special year and lays emphasis on the synthesis of modern and classical analysis at the current frontiers of knowledge.

Continuous and discrete modules are, essentially, generalizations of infective and projective modules respectively. Continuous modules provide an appropriate setting for decomposition theory of von Neumann algebras and have important applications to C*-algebras. Discrete modules constitute a dual concept and are related to number theory and algebraic geometry: they possess perfect decomposition properties. The advantage of both types of module is that the Krull-Schmidt theorem can be applied, in part, to them. The authors present here a complete account of the subject and at the same time give a unified picture of the theory. The treatment is essentially self-contained, with background facts being summarized in the first chapter. This book will be useful therefore either to individuals beginning research, or the more experienced worker in algebra and representation theory.

The theory of Lie groups is a very active part of mathematics and it is the twofold aim of these notes to provide a self-contained introduction to the subject and to make results about the structure of Lie groups and compact groups available to a wide audience. Particular emphasis is placed upon results and techniques which explicate the interplay between a Lie group and its Lie algebra, and, in keeping with current trends, a coordinate-free notation is used. Much of the general theory is illustrated by examples and exercises involving specific Lie groups.

These lecture notes begin with an introduction to topological groups and proceed to a proof of the important Pontryagin-van Kampen duality theorem and a detailed exposition of the structure of locally compact abelian groups. Measure theory and Banach algebra are entirely avoided and only a small amount of group theory and topology is required, dealing with the subject in an elementary fashion. With about a hundred exercises for the student, it is a suitable text for first-year graduate courses.

This book is based on a set of lectures given to a mixed audience of physicists and mathematicians. The desire to be intelligible to both groups is the underlying preoccupation of the author. Physicists nowadays are particularly interested in phase transitions. The typical situation is that a system of interacting particles exhibits an abrupt change of behaviour at a certain temperature, although the local forces between the particles are thought to be smooth functions of temperature. This account discusses the theory behind a simple model of such phenomena. An important tool is the mathematical discipline known as the Theory of Graphs. There are five chapters, each subdivided into sections. The first chapter is intended as a broad introduction to the subject, and it is written in a more informal manner than the rest. Notes and references for each chapter are given at the end of the chapter.

The subject of this book is the action of permutation groups on sets associated with combinatorial structures. Each chapter deals with a particular structure: groups, geometries, designs, graphs and maps respectively. A unifying theme for the first four chapters is the construction of finite simple groups. In the fifth chapter, a theory of maps on orientable surfaces is developed within a combinatorial framework. This simplifies and extends the existing literature in the field. The book is designed both as a course text and as a reference book for advanced undergraduate and graduate students. A feature is the set of carefully constructed projects, intended to give the reader a deeper understanding of the subject.

Combinatorics is an active field of mathematical study and the British Combinatorial Conference, held biennially, aims to survey the most important developments by inviting distinguished mathematicians to lecture at the meeting.

These notes comprise an introduction to combinatorial group theory and represent an extensive revision of the author's earlier book in this series, which arose from lectures to final-year undergraduates and first-year graduates at the University of Nottingham. Many new examples and exercises have been added and the treatment of a number of topics has been improved and expanded. In addition, there are new chapters on the triangle groups, small cancellation theory and groups from topology. The connections between the theory of group presentations and other areas of mathematics are emphasized throughout. The book can be used as a text for beginning research students and, for specialists in other fields, serves as an introduction both to the subject and to more advanced treatises.

A 'soliton' is a localized nonlinear wave of permanent form which may interact strongly with other solitons so that when they separate after the interaction they regain their original forms. This textbook is an account of the theory of solitons and of the diverse applications of the theory to nonlinear systems arising in the physical sciences. The essence of the book is an introduction to the method of inverse scattering. Solitary waves, cnoidal waves, conservation laws, the initial-value problem for the Korteweg-de Vries equation, the Lax method, the sine-Gordon equation and Backlund transformations are treated. The book will be useful for research workers who wish to learn about solitons as well as graduate students in mathematics, physics and engineering.

This book presents a theory motivated by the spaces LP, 0 p < l. These spaces are not locally convex, so the methods usually encountered in linear analysis (particularly the Hahn-Banach theorem) do not apply here. Questions about the size of the dual space are especially important in the non-locally convex setting, and are a central theme. Several of the classical problems in the area have been settled in the last decade, and a number of their solutions are presented here. The book begins with concrete examples (lp, LP, L0, HP) before going on to general results and important counterexamples. An F-space sampler will be of interest to research mathematicians and graduate students in functional analysis.

The purpose of these notes is to give a geometrical treatment of generalized homology and cohomology theories. The central idea is that of a 'mock bundle', which is the geometric cocycle of a general cobordism theory, and the main new result is that any homology theory is a generalized bordism theory. The book will interest mathematicians working in both piecewise linear and algebraic topology especially homology theory as it reaches the frontiers of current research in the topic. The book is also suitable for use as a graduate course in homology theory.

Methods of non-equilibrium statistical mechanics play an increasingly important role in modern turbulence research, yet the range of relevant tools and methods is so wide and developing so fast that until now there has not been a single book covering the subject. As an introduction to modern methods of statistical mechanics in turbulence, this volume rectifies that situation. The book comprises three harmonised lecture courses by world class experts in statistical physics and turbulence: John Cardy introduces Field Theory and Non-Equilibrium Statistical Mechanics; Gregory Falkovich discusses Turbulence Theory as part of Statistical Physics; and Krzysztof Gawedzki examines Soluble Models of Turbulent Transport. To encourage readers to deepen their understanding of the theoretical material, each chapter contains exercises with solutions. Essential reading for students and researchers in the field of theoretical turbulence, this volume will also interest any scientist or engineer who applies knowledge of turbulence and non-equilibrium physics to their work.

Presenting important trends in the field of stochastic analysis, this collection of thirteen articles provides an overview of recent developments and new results. Leading experts discuss a wide range of topics, ranging from an alternative set-up of rigorous probability to sampling of conditioned diffusions, from Feynmann formulas to genetic inference.

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.

This book comprises five expository articles and two research papers on topics of current interest in set theory and the foundations of mathematics. Articles by Baumgartner and Devlin introduce the reader to proper forcing.

The first half of the book is a general study of homomorphisms to simple artinian rings; the techniques developed here should be of interest to many algebraists. The second half is a more detailed study of special types of skew fields which have arisen from the work of P. M. Cohn and the author. A number of questions are settled; a version of the Jacobian conjecture for free algebras is proved and there are examples of skew field extensions of different but finite left and right dimension.

This book is a self-contained account of the theory of classgroups of group rings. The guiding philosophy has been to describe all the basic properties of such classgroups in terms of character functions. This point of view is due to A. Frohlich and it achieves a considerable simplification and clarity over previous techniques.

This collection of papers is dedicated to David Kendall, the topics will interest postgraduate and research mathematicians.

This is the expanded notes of a course intended to introduce students specializing in mathematics to some of the central ideas of traditional economics. The book should be readily accessible to anyone with some training in university mathematics; more advanced mathematical tools are explained in the appendices. Thus this text could be used for undergraduate mathematics courses or as supplementary reading for students of mathematical economics.

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.

This book is concerned with subgroups of groups of the form GL(n,D) for some division ring D. In it the authors bring together many of the advances in the theory of skew linear groups. Some aspects of skew linear groups are similar to those for linear groups, however there are often significant differences either in the method of proof or the results themselves. Topics covered in this volume include irreducibility, unipotence, locally finite-dimensional division algebras, and division algebras associated with polycyclic groups. Both authors are experts in this area of current interest in group theory, and algebraists and research students will find this an accessible account of the subject.

This book is a self-contained account of the one- and two-dimensional van der Corput method and its use in estimating exponential sums. These arise in many problems in analytic number theory. It is the first cohesive account of much of this material and will be welcomed by graduates and professionals in analytic number theory. The authors show how the method can be applied to problems such as upper bounds for the Riemann-Zeta function. the Dirichlet divisor problem, the distribution of square free numbers, and the Piatetski-Shapiro prime number theorem.

Dynamical systems is an area of intense research activity and one which finds application in many other areas of mathematics. This volume comprises a collection of survey articles that review several different areas of research. Each paper is intended to provide both an overview of a specific area and an introduction to new ideas and techniques. The authors have been encouraged to include a selection of open questions as a spur to further research. Topics covered include global bifurcations in chaotic o.d.e.s, knotted orbits in differential equations, bifurcations with symmetry, renormalization and universality, and one-dimensional dynamics. Articles include comprehensive lists of references to the research literature and consequently the volume will provide an excellent guide to dynamical systems research for graduate students coming to the subject and for research mathematicians.

The theory of Markov Processes has become a powerful tool in partial differential equations and potential theory with important applications to physics. Professor Dynkin has made many profound contributions to the subject and in this volume are collected several of his most important expository and survey articles. The content of these articles has not been covered in any monograph as yet. This account is accessible to graduate students in mathematics and operations research and will be welcomed by all those interested in stochastic processes and their applications.

Eleven of the fourteen invited speakers at a symposium held by the Oxford Mathematical Institute in June 1972 have revised their contributions and submitted them for publication in this volume. The present papers do not necessarily closely correspond with the original talks, as it was the intention of the volume editor to make this book of mathematical rather than historical interest.

The study of abelian manifolds forms a natural generalization of the theory of elliptic functions, that is, of doubly periodic functions of one complex variable. When an abelian manifold is embedded in a projective space it is termed an abelian variety in an algebraic geometrical sense. This introduction presupposes little more than a basic course in complex variables. The notes contain all the material on abelian manifolds needed for application to geometry and number theory, although they do not contain an exposition of either application. Some geometrical results are included however.

Graduate students in many branches of mathematics need to know something about topological groups and the Haar integral to enable them to understand applications in their own fields. In this introduction to the subject, Professor Higgins covers the basic theorems they are likely to need, assuming only some elementary group theory. The book is based on lecture courses given for the London M.Sc. degree in 1969 and 1972, and the treatment is more algebraic than usual, reflecting the interests of the author and his audience. The volume ends with an informal account of one important application of the Haar integral, to the representation theory of compact groups, and suggests further reading on this and similar topics.

Axiomatic set theory is the concern of this book. More particularly, the authors prove results about the coding of models M, of Zermelo-Fraenkel set theory together with the Generalized Continuum Hypothesis by using a class 'forcing' construction. By this method they extend M to another model L[a] with the same properties. L[a] is Godels universe of 'constructible' sets L, together with a set of integers a which code all the cardinality and cofinality structure of M. Some applications are also considered. Graduate students and research workers in set theory and logic will be especially interested by this account.

Professor Arnold is a prolific and versatile mathematician who has done striking work in differential equations and geometrical aspects of analysis. In this volume are collected seven of his survey articles from Russian Mathematical Surveys on singularity theory, the area to which he has made most contribution. These surveys contain Arnold's own analysis and synthesis of a decade's work. All those interested in singularity theory will find this an invaluable compilation. Professor C. T. C. Wall has written an introduction outlining the significance and content of the articles.