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This series consists of textbooks aimed at advanced undergraduates or beginning graduate students. It covers the whole range of pure mathematics, as well as topics in applied mathematics and mathematical physics that involve a substantial use of modern mathematical methods. Topics not of a standard nature though of current interest are also covered by volumes in this series.

Local Fields

J. W. S. Cassels

J. W. S. Cassels

London Mathematical Society Student Texts (No. 3)

**Print Publication Year:** 1986

**Print ISBN:** 9780521304849

**Online Publication Date:** June 2012

**Online ISBN:** 9781139171885

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139171885

The p-adic numbers, the earliest of local fields, were introduced by Hensel some 70 years ago as a natural tool in algebra number theory. Today the use of this and other local fields pervades much of mathematics, yet these simple and natural concepts, which often provide remarkably easy solutions to complex problems, are not as familiar as they should be. This book, based on postgraduate lectures at Cambridge, is meant to rectify this situation by providing a fairly elementary and self-contained introduction to local fields. After a general introduction, attention centres on the p-adic numbers and their use in number theory. There follow chapters on algebraic number theory, diophantine equations and on the analysis of a p-adic variable. This book will appeal to undergraduates, and even amateurs, interested in number theory, as well as to graduate students.

An Introduction to Twistor Theory

**Second edition**
**, Second Edition**

S. A. Huggett, K. P. Tod

S. A. Huggett, K. P. Tod

London Mathematical Society Student Texts (No. 4)

**Print Publication Year:** 1994

**Print ISBN:** 9780521451574

**Online Publication Date:** January 2010

**Online ISBN:** 9780511624018

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511624018

This book is an introduction to twistor theory and modern geometrical approaches to space-time structure at the graduate or advanced undergraduate level. It will be valuable also to the physicist as an introduction to some of the mathematics that has proved useful in these areas, and to the mathematician as an example of where sheaf cohomology and complex manifold theory can be used in physics.

Lectures on Stochastic Analysis: Diffusion Theory

Daniel W. Stroock

Daniel W. Stroock

London Mathematical Society Student Texts (No. 6)

**Print Publication Year:** 1987

**Print ISBN:** 9780521333665

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623752

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623752

This book is based on a course given at Massachusetts Institute of Technology. It is intended to be a reasonably self-contained introduction to stochastic analytic techniques that can be used in the study of certain problems. The central theme is the theory of diffusions. In order to emphasize the intuitive aspects of probabilistic techniques, diffusion theory is presented as a natural generalization of the flow generated by a vector field. Essential to the development of this idea is the introduction of martingales and the formulation of diffusion theory in terms of martingales. The book will make valuable reading for advanced students in probability theory and analysis and will be welcomed as a concise account of the subject by research workers in these fields.

Summing and Nuclear Norms in Banach Space Theory

G. J. O. Jameson

G. J. O. Jameson

London Mathematical Society Student Texts (No. 8)

**Print Publication Year:** 1987

**Print ISBN:** 9780521341349

**Online Publication Date:** October 2009

**Online ISBN:** 9780511569166

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511569166

This textbook is an introduction to the techniques of summing and nuclear norms. The author's aim is to present a clear and simple account of these ideas and to demonstrate the power of their application to a variety of Banach space questions. The style is expository and the only prerequisite is a beginner's course on Wormed linear spaces and a minimal knowledge of functional analysis. Thus, Dr Jameson is able to concentrate on important, central results and gives concrete and largely non-technical proofs, often supplying alternative proofs which both contribute something to the understanding. Final-year undergraduates and postgraduates in functional analysis will enjoy this introduction to the subject, and there are many examples and exercises throughout the text to help the reader and to demonstrate the range of application these techniques find. A list of references indicates the way for further reading.

Automorphisms of Surfaces after Nielsen and Thurston

Andrew J. Casson, Steven A. Bleiler

Andrew J. Casson, Steven A. Bleiler

London Mathematical Society Student Texts (No. 9)

**Print Publication Year:** 1988

**Print ISBN:** 9780521342032

**Online Publication Date:** August 2012

**Online ISBN:** 9780511623912

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623912

This book, which grew out of Steven Bleiler's lecture notes from a course given by Andrew Casson at the University of Texas, is designed to serve as an introduction to the applications of hyperbolic geometry to low dimensional topology. In particular it provides a concise exposition of the work of Neilsen and Thurston on the automorphisms of surfaces. The reader requires only an understanding of basic topology and linear algebra, while the early chapters on hyperbolic geometry and geometric structures on surfaces can profitably be read by anyone with a knowledge of standard Euclidean geometry desiring to learn more abour other 'geometric structures'.

Nonstandard Analysis and its Applications

Nigel Cutland

Nigel Cutland

London Mathematical Society Student Texts (No. 10)

**Print Publication Year:** 1988

**Print ISBN:** 9780521351096

**Online Publication Date:** June 2012

**Online ISBN:** 9781139172110

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139172110

This textbook is an introduction to non-standard analysis and to its many applications. Non standard analysis (NSA) is a subject of great research interest both in its own right and as a tool for answering questions in subjects such as functional analysis, probability, mathematical physics and topology. The book arises from a conference held in July 1986 at the University of Hull which was designed to provide both an introduction to the subject through introductory lectures, and surveys of the state of research. The first part of the book is devoted to the introductory lectures and the second part consists of presentations of applications of NSA to dynamical systems, topology, automata and orderings on words, the non- linear Boltzmann equation and integration on non-standard hulls of vector lattices. One of the book's attractions is that a standard notation is used throughout so the underlying theory is easily applied in a number of different settings. Consequently this book will be ideal for graduate students and research mathematicians coming to the subject for the first time and it will provide an attractive and stimulating account of the subject.

Undergraduate Algebraic Geometry

Miles Reid

Miles Reid

London Mathematical Society Student Texts (No. 12)

**Print Publication Year:** 1988

**Print ISBN:** 9780521355599

**Online Publication Date:** June 2012

**Online ISBN:** 9781139163699

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139163699

Algebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. With the minimum of prerequisites, Dr. Reid introduces the reader to the basic concepts of algebraic geometry, including: plane conics, cubics and the group law, affine and projective varieties, and nonsingularity and dimension. He stresses the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book contains numerous examples and exercises illustrating the theory.

An Introduction to Hankel Operators

Jonathan R. Partington

Jonathan R. Partington

London Mathematical Society Student Texts (No. 13)

**Print Publication Year:** 1989

**Print ISBN:** 9780521366113

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623769

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623769

Hankel operators are of wide application in mathematics (functional analysis, operator theory, approximation theory) and engineering (control theory, systems analysis) and this account of them is both elementary and rigorous. The book is based on graduate lectures given to an audience of mathematicians and control engineers, but to make it reasonably self-contained, the author has included several appendices on mathematical topics unlikely to be met by undergraduate engineers. The main prerequisites are basic complex analysis and some functional analysis, but the presentation is kept straightforward, avoiding unnecessary technicalities so that the fundamental results and their applications are evident. Some 45 exercises are included.

Combinatorial Group Theory

**A Topological Approach**

Daniel E. Cohen

Daniel E. Cohen

London Mathematical Society Student Texts (No. 14)

**Print Publication Year:** 1989

**Print ISBN:** 9780521341332

**Online Publication Date:** January 2010

**Online ISBN:** 9780511565878

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511565878

In this book, developed from courses taught at the University of London, the author aims to show the value of using topological methods in combinatorial group theory. The topological material is given in terms of the fundamental groupoid, giving results and proofs that are both stronger and simpler than the traditional ones. Several chapters deal with covering spaces and complexes, an important method, which is then applied to yield the major Schreier and Kurosh subgroup theorems. The author presents a full account of Bass-Serre theory and discusses the word problem, in particular, its unsolvability and the Higman Embedding Theorem. Included for completeness are the relevant results of computability theory.

Presentations of Groups

**Second edition**

D. L. Johnson

D. L. Johnson

London Mathematical Society Student Texts (No. 15)

**Print Publication Year:** 1997

**Print ISBN:** 9780521585422

**Online Publication Date:** June 2012

**Online ISBN:** 9781139168410

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139168410

Emphasizing computational techniques, this book provides an accessible and lucid introduction to combinatorial group theory. Rigorous proofs of all theorems and a light, informal style make Presentations of Groups a self-contained combinatorics class. Numerous and diverse exercises provide readers with a thorough overview of the subject. While catering to combinatorics beginners, this book also includes the frontiers of research, and explains software packages such as GAP, MAGMA, and QUOTPIC. This new edition has been revised throughout, including new exercises and an additional chapter on proving certain groups are infinite. Aimed at advanced undergraduates, this book will be a resource for graduate students and researchers.

Aspects of Quantum Field Theory in Curved Spacetime

Stephen A. Fulling

Stephen A. Fulling

London Mathematical Society Student Texts (No. 17)

**Print Publication Year:** 1989

**Print ISBN:** 9780521344005

**Online Publication Date:** June 2012

**Online ISBN:** 9781139172073

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139172073

This introduction to the theory of quantum fields in curved spacetime, intended for mathematicians, arose from a course taught to graduate students and is designed for self-study or advanced courses in relativity and quantum field theory. The style is informal and some knowledge of general relativity and differential geometry is assumed, yet the author does supply background material on function analysis and quantum field theory as required. Physicists should also gain a sound grasp of various aspects of the theory, some of which have not been particularly emphasized in the existing review literature.

Braids and Coverings

**Selected Topics**

Vagn Lundsgaard Hansen

Vagn Lundsgaard Hansen

London Mathematical Society Student Texts (No. 18)

**Print Publication Year:** 1989

**Print ISBN:** 9780521384797

**Online Publication Date:** January 2010

**Online ISBN:** 9780511613098

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511613098

This book is based on a graduate course taught by the author at the University of Maryland. The lecture notes have been revised and augmented by examples. The first two chapters develop the elementary theory of Artin Braid groups, both geometrically and via homotopy theory, and discuss the link between knot theory and the combinatorics of braid groups through Markou's Theorem. The final two chapters give a detailed investigation of polynomial covering maps, which may be viewed as a homomorphism of the fundamental group of the base space into the Artin Braid group on n strings.

Representations of Finite Groups of Lie Type

François Digne, Jean Michel

François Digne, Jean Michel

London Mathematical Society Student Texts (No. 21)

**Print Publication Year:** 1991

**Print ISBN:** 9780521401173

**Online Publication Date:** June 2012

**Online ISBN:** 9781139172417

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139172417

The authors aim to treat the basic theory of representations of finite groups of Lie type, such as linear, unitary, orthogonal and symplectic groups. They emphasize the Curtis-Alvis duality map and Mackey's theorem and the results that can be deduced from it. They also discuss Deligne-Lusztig induction. This will be the first elementary treatment of this material in book form and will be welcomed by beginning graduate students in algebra.

Designs, Graphs, Codes and their Links

P. J. Cameron, J. H. van Lint

P. J. Cameron, J. H. van Lint

London Mathematical Society Student Texts (No. 22)

**Print Publication Year:** 1991

**Print ISBN:** 9780521413251

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623714

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623714

This book stresses the connection between, and the applications of, design theory to graphs and codes. Beginning with a brief introduction to design theory and the necessary background, the book also provides relevant topics for discussion from the theory of graphs and codes.

Complex Algebraic Curves

Frances Kirwan

Frances Kirwan

London Mathematical Society Student Texts (No. 23)

**Print Publication Year:** 1992

**Print ISBN:** 9780521412513

**Online Publication Date:** June 2012

**Online ISBN:** 9780511623929

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623929

This development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics. They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis.

LMSST: 24 Lectures on Elliptic Curves

J. W. S. Cassels

J. W. S. Cassels

London Mathematical Society Student Texts (No. 24)

**Print Publication Year:** 1991

**Print ISBN:** 9780521415170

**Online Publication Date:** June 2012

**Online ISBN:** 9781139172530

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139172530

The study of special cases of elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centers of research in number theory. This book, addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. The central portion deals with curves over the rationals: the Mordell-Wei finite basis theorem, points of finite order (Nagell-Lutz), etc. The treatment is structured by the local-global standpoint and culminates in the description of the Tate-Shafarevich group as the obstruction to a Hasse principle. In an introductory section the Hasse principle for conics is discussed. The book closes with sections on the theory over finite fields (the "Riemann hypothesis for function fields") and recently developed uses of elliptic curves for factoring large integers. Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no knowledge either of algebraic number theory or algebraic geometry is needed. The p-adic numbers are introduced from scratch. Many examples and exercises are included for the reader, and those new to elliptic curves, whether they are graduate students or specialists from other fields, will find this a valuable introduction.

Hyperbolic Geometry

Birger Iversen

Birger Iversen

London Mathematical Society Student Texts (No. 25)

**Print Publication Year:** 1992

**Print ISBN:** 9780521435086

**Online Publication Date:** October 2009

**Online ISBN:** 9780511569333

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511569333

Although it arose from purely theoretical considerations of the underlying axioms of geometry, the work of Einstein and Dirac has demonstrated that hyperbolic geometry is a fundamental aspect of modern physics. In this book, the rich geometry of the hyperbolic plane is studied in detail, leading to the focal point of the book, Poincare's polygon theorem and the relationship between hyperbolic geometries and discrete groups of isometries. Hyperbolic 3-space is also discussed, and the directions that current research in this field is taking are sketched. This will be an excellent introduction to hyperbolic geometry for students new to the subject, and for experts in other fields.

Elementary Theory of L-functions and Eisenstein Series

Haruzo Hida

Haruzo Hida

London Mathematical Society Student Texts (No. 26)

**Print Publication Year:** 1993

**Print ISBN:** 9780521434119

**Online Publication Date:** April 2010

**Online ISBN:** 9780511623691

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623691

This book is a comprehensive and systematic account of the theory of p-adic and classical modular forms and the theory of the special values of arithmetic L-functions and p-adic L-functions. The approach is basically algebraic, and the treatment is elementary. No deep knowledge from algebraic geometry and representation theory is required. The author's main tool in dealing with these problems is taken from cohomology theory over Riemann surfaces, which is also explained in detail in the book. He also gives a concise but thorough treatment of analytic continuation and functional equation. Graduate students wishing to know more about L-functions will find this a unique introduction to this fascinating branch of mathematics.

Hilbert Space

**Compact Operators and the Trace Theorem**

J. R. Retherford

J. R. Retherford

London Mathematical Society Student Texts (No. 27)

**Print Publication Year:** 1993

**Print ISBN:** 9780521418843

**Online Publication Date:** June 2012

**Online ISBN:** 9781139172592

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139172592

The aim of this book is to provide the reader with a virtually self-contained treatment of Hilbert space theory leading to an elementary proof of the Lidskij trace theorem. The author assumes the reader is familiar with linear algebra and advanced calculus, and develops everything needed to introduce the ideas of compact, self-adjoint, Hilbert-Schmidt and trace class operators. Many exercises and hints are included, and throughout the emphasis is on a user-friendly approach.

Potential Theory in the Complex Plane

Thomas Ransford

Thomas Ransford

London Mathematical Society Student Texts (No. 28)

**Print Publication Year:** 1995

**Print ISBN:** 9780521461207

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623776

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623776

Ransford provides an introduction to the subject, concentrating on the important case of two dimensions, and emphasizing its links with complex analysis. This is reflected in the large number of applications, which include Picard's theorem, the Phragmén-Lindelöf principle, the Radó-Stout theorem, Lindelöf's theory of asymptotic values, the Riemann mapping theorem (including continuity at the boundary), the Koebe one-quarter theorem, Hilbert's lemniscate theorem, and the sharp quantitative form of Runge's theorem. In addition, there is a chapter on connections with functional analysis and dynamical systems, which shows how the theory can be applied to other parts of mathematics and gives a flavor of some recent research in the area.

Undergraduate Commutative Algebra

Miles Reid

Miles Reid

London Mathematical Society Student Texts (No. 29)

**Print Publication Year:** 1995

**Print ISBN:** 9780521452557

**Online Publication Date:** June 2012

**Online ISBN:** 9781139172721

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139172721

In this well-written introduction to commutative algebra, the author shows the link between commutative ring theory and algebraic geometry. In addition to standard material, the book contrasts the methods and ideology of modern abstract algebra with concrete applications in algebraic geometry and number theory. Professor Reid begins with a discussion of modules and Noetherian rings before moving on to finite extensions and the Noether normalization. Sections on the nullstellensatz and rings of fractions precede sections on primary decomposition and normal integral domains. This book is ideal for anyone seeking a primer on commutative algebra.

The Laplacian on a Riemannian Manifold

**An Introduction to Analysis on Manifolds**

Steven Rosenberg

Steven Rosenberg

London Mathematical Society Student Texts (No. 31)

**Print Publication Year:** 1997

**Print ISBN:** 9780521463003

**Online Publication Date:** December 2009

**Online ISBN:** 9780511623783

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623783

This text on analysis on Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. The main theme is the study of heat flow associated to the Laplacians on differential forms. This provides a unified treatment of Hodge theory and the supersymmetric proof of the Chern-Gauss-Bonnet theorem. In particular, there is a careful treatment of the heat kernel for the Laplacian on functions. The author develops the Atiyah-Singer index theorem and its applications (without complete proofs) via the heat equation method. Rosenberg also treats zeta functions for Laplacians and analytic torsion, and lays out the recently uncovered relation between index theory and analytic torsion. The text is aimed at students who have had a first course in differentiable manifolds, and the author develops the Riemannian geometry used from the beginning. There are over 100 exercises with hints.

Lectures on Lie Groups and Lie Algebras

Roger W. Carter, Ian G. MacDonald, Graeme B. Segal, Foreword by M. Taylor

Roger W. Carter, Ian G. MacDonald, Graeme B. Segal, Foreword by M. Taylor

London Mathematical Society Student Texts (No. 32)

**Print Publication Year:** 1995

**Print ISBN:** 9780521495790

**Online Publication Date:** June 2012

**Online ISBN:** 9781139172882

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139172882

Three of the leading figures in the field have composed this excellent introduction to the theory of Lie groups and Lie algebras. Together these lectures provide an elementary account of the theory that is unsurpassed. In the first part, Roger Carter concentrates on Lie algebras and root systems. In the second Graeme Segal discusses Lie groups. And in the final part, Ian Macdonald gives an introduction to special linear groups. Graduate students requiring an introduction to the theory of Lie groups and their applications should look no further than this book.

A Primer of Algebraic D-Modules

S. C. Coutinho

S. C. Coutinho

London Mathematical Society Student Texts (No. 33)

**Print Publication Year:** 1995

**Print ISBN:** 9780521551199

**Online Publication Date:** December 2009

**Online ISBN:** 9780511623653

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623653

The theory of D-modules is a rich area of study combining ideas from algebra and differential equations, and it has significant applications to diverse areas such as singularity theory and representation theory. This book introduces D-modules and their applications, avoiding all unnecessary technicalities. The author takes an algebraic approach, concentrating on the role of the Weyl algebra. The author assumes very few prerequisites, and the book is virtually self-contained. The author includes exercises at the end of each chapter and gives the reader ample references to the more advanced literature. This is an excellent introduction to D-modules for all who are new to this area.

Complex Algebraic Surfaces

**Second edition**

Arnaud Beauville

Arnaud Beauville

London Mathematical Society Student Texts (No. 34)

**Print Publication Year:** 1996

**Print ISBN:** 9780521495103

**Online Publication Date:** March 2010

**Online ISBN:** 9780511623936

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623936

The classification of algebraic surfaces is an intricate and fascinating branch of mathematics, developed over more than a century and still an active area of research today. In this book, Professor Beauville gives a lucid and concise account of the subject, expressed simply in the language of modern topology and sheaf theory, and accessible to any budding geometer. A chapter on preliminary material ensures that this volume is self-contained while the exercises succeed both in giving the flavor of the classical subject, and in equipping the reader with the techniques needed for research. The book is aimed at graduate students in geometry and topology.

Young Tableaux

**With Applications to Representation Theory and Geometry**

William Fulton

William Fulton

London Mathematical Society Student Texts (No. 35)

**Print Publication Year:** 1996

**Print ISBN:** 9780521561440

**Online Publication Date:** June 2012

**Online ISBN:** 9780511626241

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511626241

This book develops the combinatorics of Young tableaux and shows them in action in the algebra of symmetric functions, representations of the symmetric and general linear groups, and the geometry of flag varieties. The first part of the book is a self-contained presentation of the basic combinatorics of Young tableaux, including the remarkable constructions of "bumping" and "sliding", and several interesting correspondences. In Part II the author uses these results to study representations with geometry on Grassmannians and flag manifolds, including their Schubert subvarieties, and the related Schubert polynomials. Much of this material has never before appeared in book form. There are numerous exercises throughout, with hints and answers provided. Researchers in representation theory and algebraic geometry as well as in combinatorics will find this book interesting and useful, while students will find the intuitive presentation easy to follow.

A Mathematical Introduction to Wavelets

P. Wojtaszczyk

P. Wojtaszczyk

London Mathematical Society Student Texts (No. 37)

**Print Publication Year:** 1997

**Print ISBN:** 9780521570206

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623790

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623790

This book presents a mathematical introduction to the theory of orthogonal wavelets and their uses in analyzing functions and function spaces, both in one and in several variables. Starting with a detailed and self-contained discussion of the general construction of one dimensional wavelets from multiresolution analysis, the book presents in detail the most important wavelets: spline wavelets, Meyer's wavelets and wavelets with compact support. It then moves to the corresponding multivariable theory and gives genuine multivariable examples. The author discusses wavelet decompositions in Lp spaces, Hardy spaces and Besov spaces and provides wavelet characterizations of those spaces. Also included are periodic wavelets or wavelets not associated with a multiresolution analysis. This will be an invaluable book for those wishing to learn about the mathematical foundations of wavelets.

Harmonic Maps, Loop Groups, and Integrable Systems

Martin A. Guest

Martin A. Guest

London Mathematical Society Student Texts (No. 38)

**Print Publication Year:** 1997

**Print ISBN:** 9780521580854

**Online Publication Date:** June 2012

**Online ISBN:** 9781139174848

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139174848

This is an accessible introduction to some of the fundamental connections among differential geometry, Lie groups, and integrable Hamiltonian systems. The text demonstrates how the theory of loop groups can be used to study harmonic maps. By concentrating on the main ideas and examples, the author leads up to topics of current research. The book is suitable for students who are beginning to study manifolds and Lie groups, and should be of interest both to mathematicians and to theoretical physicists as well.

Set Theory for the Working Mathematician

Krzysztof Ciesielski

Krzysztof Ciesielski

London Mathematical Society Student Texts (No. 39)

**Print Publication Year:** 1997

**Print ISBN:** 9780521594417

**Online Publication Date:** June 2012

**Online ISBN:** 9781139173131

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139173131

This text presents methods of modern set theory as tools that can be usefully applied to other areas of mathematics. The author describes numerous applications in abstract geometry and real analysis and, in some cases, in topology and algebra. The book begins with a tour of the basics of set theory, culminating in a proof of Zorn's Lemma and a discussion of some of its applications. The author then develops the notions of transfinite induction and descriptive set theory, with applications to the theory of real functions. The final part of the book presents the tools of "modern" set theory: Martin's Axiom, the Diamond Principle, and elements of forcing. Written primarily as a text for beginning graduate or advanced level undergraduate students, this book should also interest researchers wanting to learn more about set theoretical techniques applicable to their fields.

Dynamical Systems and Ergodic Theory

Mark Pollicott, Michiko Yuri

Mark Pollicott, Michiko Yuri

London Mathematical Society Student Texts (No. 40)

**Print Publication Year:** 1998

**Print ISBN:** 9780521572941

**Online Publication Date:** March 2015

**Online ISBN:** 9781139173049

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139173049

This book is an introduction to topological dynamics and ergodic theory. It is divided into a number of relatively short chapters with the intention that each may be used as a component of a lecture course tailored to the particular audience. The authors provide a number of applications, principally to number theory and arithmetic progressions (through Van der Waerden's theorem and Szemerdi's theorem). This text is suitable for advanced undergraduate and beginning graduate students.

The Algorithmic Resolution of Diophantine Equations

**A Computational Cookbook**

Nigel P. Smart

Nigel P. Smart

London Mathematical Society Student Texts (No. 41)

**Print Publication Year:** 1998

**Print ISBN:** 9780521641562

**Online Publication Date:** May 2013

**Online ISBN:** 9781107359994

**Book DOI:** http://dx.doi.org/10.1017/CBO9781107359994

Beginning with a brief introduction to algorithms and diophantine equations, this volume provides a coherent modern account of the methods used to find all the solutions to certain diophantine equations, particularly those developed for use on a computer. The study is divided into three parts, emphasizing approaches with a wide range of applications. The first section considers basic techniques including local methods, sieving, descent arguments and the LLL algorithm. The second section explores problems that can be solved using Baker's theory of linear forms in logarithms. The final section looks at problems associated with curves, focusing on rational and integral points on elliptic curves. Each chapter concludes with a useful set of exercises. A detailed bibliography is included. This book will appeal to graduate students and research workers interested in solving diophantine equations using computational methods.

Equilibrium States in Ergodic Theory

Gerhard Keller

Gerhard Keller

London Mathematical Society Student Texts (No. 42)

**Print Publication Year:** 1998

**Print ISBN:** 9780521594202

**Online Publication Date:** April 2013

**Online ISBN:** 9781107359987

**Book DOI:** http://dx.doi.org/10.1017/CBO9781107359987

This book provides a detailed introduction to the ergodic theory of equilibrium states giving equal weight to two of its most important applications, namely to equilibrium statistical mechanics on lattices and to (time discrete) dynamical systems. It starts with a chapter on equilibrium states on finite probability spaces that introduces the main examples for the theory on an elementary level. After two chapters on abstract ergodic theory and entropy, equilibrium states and variational principles on compact metric spaces are introduced, emphasizing their convex geometric interpretation. Stationary Gibbs measures, large deviations, the Ising model with external field, Markov measures, Sinai-Bowen-Ruelle measures for interval maps and dimension maximal measures for iterated function systems are the topics to which the general theory is applied in the last part of the book. The text is self contained except for some measure theoretic prerequisites that are listed (with references to the literature) in an appendix.

Fourier Analysis on Finite Groups and Applications

Audrey Terras

Audrey Terras

London Mathematical Society Student Texts (No. 43)

**Print Publication Year:** 1999

**Print ISBN:** 9780521451086

**Online Publication Date:** July 2010

**Online ISBN:** 9780511626265

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511626265

This book gives a friendly introduction to Fourier analysis on finite groups, both commutative and noncommutative. Aimed at students in mathematics, engineering and the physical sciences, it examines the theory of finite groups in a manner both accessible to the beginner and suitable for graduate research. With applications in chemistry, error-correcting codes, data analysis, graph theory, number theory and probability, the book presents a concrete approach to abstract group theory through applied examples, pictures and computer experiments. The author divides the book into two parts. In the first part, she parallels the development of Fourier analysis on the real line and the circle, and then moves on to analog of higher dimensional Euclidean space. The second part emphasizes matrix groups such as the Heisenberg group of upper triangular 2x2 matrices with 1's down the diagonal and entries in a finite field. The book concludes with an introduction to zeta functions on finite graphs via the trace formula.

Classical Invariant Theory

Peter J. Olver

Peter J. Olver

London Mathematical Society Student Texts (No. 44)

**Print Publication Year:** 1999

**Print ISBN:** 9780521552431

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623660

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623660

There has been a resurgence of interest in classical invariant theory driven by several factors: new theoretical developments; a revival of computational methods coupled with powerful new computer algebra packages; and a wealth of new applications, ranging from number theory to geometry, physics to computer vision. This book provides readers with a self-contained introduction to the classical theory as well as modern developments and applications. The text concentrates on the study of binary forms (polynomials) in characteristic zero, and uses analytical as well as algebraic tools to study and classify invariants, symmetry, equivalence and canonical forms. A variety of innovations make this text of interest even to veterans of the subject; these include the use of differential operators and the transform approach to the symbolic method, extension of results to arbitrary functions, graphical methods for computing identities and Hilbert bases, complete systems of rationally and functionally independent covariants, introduction of Lie group and Lie algebra methods, as well as a new geometrical theory of moving frames and applications. Aimed at advanced undergraduate and graduate students the book includes many exercises and historical details, complete proofs of the fundamental theorems, and a lively and provocative exposition.

Permutation Groups

Peter J. Cameron

Peter J. Cameron

London Mathematical Society Student Texts (No. 45)

**Print Publication Year:** 1999

**Print ISBN:** 9780521653022

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623677

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623677

Permutation groups are one of the oldest topics in algebra. Their study has recently been revolutionized by new developments, particularly the Classification of Finite Simple Groups, but also relations with logic and combinatorics, and importantly, computer algebra systems have been introduced that can deal with large permutation groups. This text summarizes these developments, including an introduction to relevant computer algebra systems, sketch proofs of major theorems, and many examples of applying the Classification of Finite Simple Groups. It is aimed at beginning graduate students and experts in other areas, and grew from a short course at the EIDMA institute in Eindhoven.

Introductory Lectures on Rings and Modules

John A. Beachy

John A. Beachy

London Mathematical Society Student Texts (No. 47)

**Print Publication Year:** 1999

**Print ISBN:** 9780521643405

**Online Publication Date:** June 2012

**Online ISBN:** 9781139173315

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139173315

The focus of this book is the study of the noncommutative aspects of rings and modules, and the style will make it accessible to anyone with a background in basic abstract algebra. Features of interest include an early introduction of projective and injective modules; a module theoretic approach to the Jacobson radical and the Artin-Wedderburn theorem; the use of Baer's criterion for injectivity to prove the structure theorem for finitely generated modules over a principal ideal domain; and applications of the general theory to the representation theory of finite groups. Optional material includes a section on modules over the Weyl algebras and a section on Goldie's theorem. When compared to other more encyclopedic texts, the sharp focus of this book accommodates students meeting this material for the first time. It can be used as a first-year graduate text or as a reference for advanced undergraduates.

Set Theory

Andras Hajnal, Peter Hamburger, Translated by Attila Mate

Andras Hajnal, Peter Hamburger, Translated by Attila Mate

London Mathematical Society Student Texts (No. 48)

**Print Publication Year:** 1999

**Print ISBN:** 9780521593441

**Online Publication Date:** May 2010

**Online ISBN:** 9780511623561

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623561

This is a classic introduction to set theory, suitable for students with no previous knowledge of the subject. Providing complete, up-to-date coverage, the book is based in large part on courses given over many years by Professor Hajnal. The first part introduces all the standard notions of the subject; the second part concentrates on combinatorial set theory. Exercises are included throughout and a new section of hints has been added to assist the reader.

An Introduction to K-Theory for C*-Algebras

M. Rørdam, F. Larsen, N. Laustsen

M. Rørdam, F. Larsen, N. Laustsen

London Mathematical Society Student Texts (No. 49)

**Print Publication Year:** 2000

**Print ISBN:** 9780521783347

**Online Publication Date:** December 2009

**Online ISBN:** 9780511623806

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623806

Over the past twenty-five years K-theory has become an integrated part of the study of C*-algebras. This book gives a very elementary introduction to this interesting and rapidly growing area of mathematics. The authors cover the basic properties of the functors K and K1 and their interrelationship. In particular, the Bott periodicity theorem is proved (Atiyah's proof), and the six-term exact sequence is derived. The theory is well illustrated with 120 exercises and examples, making the book ideal for beginning graduate students in functional analysis, especially operator algebras, and for researchers from other areas of mathematics who want to learn about this subject.

A Brief Guide to Algebraic Number Theory

H. P. F. Swinnerton-Dyer

H. P. F. Swinnerton-Dyer

London Mathematical Society Student Texts (No. 50)

**Print Publication Year:** 2001

**Print ISBN:** 9780521802925

**Online Publication Date:** June 2012

**Online ISBN:** 9781139173360

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139173360

This account of Algebraic Number Theory is written primarily for beginning graduate students in pure mathematics, and encompasses everything that most such students are likely to need; others who need the material will also find it accessible. It assumes no prior knowledge of the subject, but a firm basis in the theory of field extensions at an undergraduate level is required, and an appendix covers other prerequisites. The book covers the two basic methods of approaching Algebraic Number Theory, using ideals and valuations, and includes material on the most usual kinds of algebraic number field, the functional equation of the zeta function and a substantial digression on the classical approach to Fermat's Last Theorem, as well as a comprehensive account of class field theory. Many exercises and an annotated reading list are also included.

Steps in Commutative Algebra

**Second edition**

Rodney Y. Sharp

Rodney Y. Sharp

London Mathematical Society Student Texts (No. 51)

**Print Publication Year:** 2001

**Print ISBN:** 9780521646239

**Online Publication Date:** January 2010

**Online ISBN:** 9780511623684

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511623684

This introductory account of commutative algebra is aimed at students with a background only in basic algebra. Professor Sharp's book provides a good foundation from which the reader can proceed to more advanced works in commutative algebra or algebraic geometry. This new edition contains additional chapters on regular sequences and on Cohen-Macaulay rings.

Finite Markov Chains and Algorithmic Applications

Olle Häggström

Olle Häggström

London Mathematical Society Student Texts (No. 52)

**Print Publication Year:** 2002

**Print ISBN:** 9780521813570

**Online Publication Date:** March 2010

**Online ISBN:** 9780511613586

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511613586

This text is ideal for advanced undergraduate or beginning graduate students. The author first develops the necessary background in probability theory and Markov chains before using it to study a range of randomized algorithms with important applications in optimization and other problems in computing. The book will appeal not only to mathematicians, but to students of computer science who will discover much useful material. This clear and concise introduction to the subject has numerous exercises that will help students to deepen their understanding.

The Prime Number Theorem

G. J. O. Jameson

G. J. O. Jameson

London Mathematical Society Student Texts (No. 53)

**Print Publication Year:** 2003

**Print ISBN:** 9780521814119

**Online Publication Date:** June 2012

**Online ISBN:** 9781139164986

**Book DOI:** http://dx.doi.org/10.1017/CBO9781139164986

At first glance the prime numbers appear to be distributed in a very irregular way amongst the integers, but it is possible to produce a simple formula that tells (in an approximate but well-defined sense) how many primes can be found that are less than any integer. The prime number theorem tells what this formula is and it is indisputably one of the the great classical theorems of mathematics. This textbook introduces the prime number theorem and is suitable for advanced undergraduates and beginning graduate students. The author deftly shows how analytical tools can be used in number theory to attack a 'real' problem.

Elementary Number Theory, Group Theory and Ramanujan Graphs

Giuliana Davidoff, Peter Sarnak, Alain Valette

Giuliana Davidoff, Peter Sarnak, Alain Valette

London Mathematical Society Student Texts (No. 55)

**Print Publication Year:** 2003

**Print ISBN:** 9780521824262

**Online Publication Date:** December 2009

**Online ISBN:** 9780511615825

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511615825

This text is a self-contained study of expander graphs, specifically, their explicit construction. Expander graphs are highly connected but sparse, and while being of interest within combinatorics and graph theory, they can also be applied to computer science and engineering. Only a knowledge of elementary algebra, analysis and combinatorics is required because the authors provide the necessary background from graph theory, number theory, group theory and representation theory. Thus the text can be used as a brief introduction to these subjects and their synthesis in modern mathematics.

Logic, Induction and Sets

Thomas Forster

Thomas Forster

London Mathematical Society Student Texts (No. 56)

**Print Publication Year:** 2003

**Print ISBN:** 9780521826211

**Online Publication Date:** June 2012

**Online ISBN:** 9780511810282

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511810282

Philosophical considerations, which are often ignored or treated casually, are given careful consideration in this introduction. Thomas Forster places the notion of inductively defined sets (recursive datatypes) at the center of his exposition resulting in an original analysis of well established topics. The presentation illustrates difficult points and includes many exercises. Little previous knowledge of logic is required and only a knowledge of standard undergraduate mathematics is assumed.

Introduction to Banach Algebras, Operators, and Harmonic Analysis

H. Garth Dales, Pietro Aiena, Jörg Eschmeier, Kjeld Laursen, George A. Willis

H. Garth Dales, Pietro Aiena, Jörg Eschmeier, Kjeld Laursen, George A. Willis

London Mathematical Society Student Texts (No. 57)

**Print Publication Year:** 2003

**Print ISBN:** 9780521828932

**Online Publication Date:** December 2009

**Online ISBN:** 9780511615429

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511615429

Based on lectures given at an instructional course, this volume enables readers with a basic knowledge of functional analysis to access key research in the field. The authors survey several areas of current interest, making this volume ideal preparatory reading for students embarking on graduate work as well as for mathematicians working in related areas.

Computational Algebraic Geometry

Hal Schenck

Hal Schenck

London Mathematical Society Student Texts (No. 58)

**Print Publication Year:** 2003

**Print ISBN:** 9780521829649

**Online Publication Date:** July 2010

**Online ISBN:** 9780511756320

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511756320

Recent advances in computing and algorithms make it easier to do many classical problems in algebra. Suitable for graduate students, this book brings advanced algebra to life with many examples. The first three chapters provide an introduction to commutative algebra and connections to geometry. The remainder of the book focuses on three active areas of contemporary algebra: homological algebra; algebraic combinatorics and algebraic topology; and algebraic geometry.

Frobenius Algebras and 2-D Topological Quantum Field Theories

Joachim Kock

Joachim Kock

London Mathematical Society Student Texts (No. 59)

**Print Publication Year:** 2003

**Print ISBN:** 9780521832670

**Online Publication Date:** January 2010

**Online ISBN:** 9780511615443

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511615443

Describing a striking connection between topology and algebra, rather than only proving the theorem, this study demonstrates how the result fits into a more general pattern. Throughout the text emphasis is on the interplay between algebra and topology, with graphical interpretation of algebraic operations, and topological structures described algebraically in terms of generators and relations. Includes numerous exercises and examples.

Linear Operators and Linear Systems

**An Analytical Approach to Control Theory**

Jonathan R. Partington

Jonathan R. Partington

London Mathematical Society Student Texts (No. 60)

**Print Publication Year:** 2004

**Print ISBN:** 9780521837347

**Online Publication Date:** May 2010

**Online ISBN:** 9780511616693

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511616693

This volume presents an introduction to the common ground between operator theory and linear systems theory. Pure mathematical topics are included such as Hardy spaces, closed operators, the gap metric, semigroups, shift-invariant subspaces, the commutant lifting theorem and almost-periodic functions, which would be suitable for a course in functional analysis. The book also includes applications to partial differential equations, the stability and stabilization of linear systems, power signal spaces, and delay systems, treated from an input/output point of view.

An Introduction to Noncommutative Noetherian Rings

**Second edition**

K. R. Goodearl, R. B. Warfield, Jr

K. R. Goodearl, R. B. Warfield, Jr

London Mathematical Society Student Texts (No. 61)

**Print Publication Year:** 2004

**Print ISBN:** 9780521836876

**Online Publication Date:** November 2010

**Online ISBN:** 9780511841699

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511841699

This introduction to noncommutative noetherian rings, accessible to anyone with a basic background in abstract algebra, can be used as a second-year graduate text, or as a self-contained reference. Extensive explanatory material is given, and exercises are integrated throughout. New material includes the basic types of quantum groups.

Topics from One-Dimensional Dynamics

Karen M. Brucks, Henk Bruin

Karen M. Brucks, Henk Bruin

London Mathematical Society Student Texts (No. 62)

**Print Publication Year:** 2004

**Print ISBN:** 9780521838962

**Online Publication Date:** August 2012

**Online ISBN:** 9780511617171

**Book DOI:** http://dx.doi.org/10.1017/CBO9780511617171

One-dimensional dynamics has generated many results, and avenues of active mathematical research with numerous inroads to this research remain to be pursued by the advanced undergraduate or beginning graduate student. While much of the material in this book is not covered elsewhere, some aspects present new research topics whose connections are drawn to other research areas from the text. Although the material presented is not meant to be approached in a linear fashion, anybody with an interest in dynamics will find many topics of interest.

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