Bøger i London Mathematical Society Student Texts serien

The study of singularities uses techniques from algebra, algebraic geometry, complex analysis and topology. This book introduces graduate students to this attractive area of mathematics. It is based on a MSc course taught by the author and also is an original synthesis, with new views and results not found elsewhere.

One-dimensional dynamics owns many deep results and avenues of active mathematical research. This book provides glimpses into the subject, and novel connections are drawn with other research areas, with the hope that the results presented illuminate the beauty and excitement of the field. Much of this material is covered nowhere else in textbook format.

This book gives a summary of recent developments in the study of permutation groups, 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.

This book presents an introduction to the common ground between operator theory and linear systems theory. Suitable for students of functional analysis, this book also acts as an introduction to a mathematical approach to systems and control for graduate students in departments of applied mathematics or engineering.

This 2003 book proves striking results connecting topology and algebra and shows how the result fits into a more general pattern. It will prove valuable to students or researchers entering this field who will learn a host of modern techniques. There are numerous exercises and examples making the book suitable for teaching.

This is a classic introduction to set theory in three parts. The first part gives a general introduction to set theory, suitable for undergraduates; the second part gives a more formal foundation to axiomatic set theory; and the final part gives an introduction to modern tools of combinatorial set theory.

The prime number theorem is indisputably one of the great classical theorems of mathematics. Suitable for advanced undergraduates and beginning graduates, this textbook demonstrates how the tools of analysis can be used in number theory to attack a famous problem.

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. It can be used as a first-year graduate text or as a reference for advanced undergraduates.

This text concentrates on the typical methods of modern set theory: transfinite induction, Zorn's Lemma, the Continuum Hypothesis, Martin's Axiom, the Diamond Principle and elements of forcing. It supplies students and researchers with the tools most applicable to other areas of pure mathematics, such as abstract geometry, real analysis, topology, and algebra.

The theory of p-adic and classic modular forms, and the study of arithmetic and p-adic L-functions has proved to be a fruitful area of mathematics over the last decade. Professor Hida has given courses on these topics worldwide and here provides the reader with an elementary insight into the theory of L-functions.

Professor Retherford's aim in 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. Many exercises and hints are included, and throughout the emphasis is on a user-friendly approach.

This is an introduction to logic and the axiomatization of set theory from a unique standpoint. Difficult points are presented in an engaging fashion and furthered by the aid of many exercises. Little previous knowledge of logic is required, only a background of standard undergraduate mathematics is assumed.

This 2002 book is ideal for advanced undergraduate or beginning graduate students. The author first develops the necessary background in probability theory and Markov chains before applying it to study a range of randomised algorithms with important applications in optimisation and other problems in computing.

This book is based on a graduate course taught by the author at the University of Maryland. The work falls into two strands: first the elementary theory of Artin Braid groups is developed and the link between knot theory and the combinatorics of braid groups is discussed through Markov's theorem; this is followed by an investigation of polynomial maps.

The book is a self-contained introduction to the results and computational methods in classical invariant theory. It relies on minimal mathematical prerequisites, and can be read by advanced undergraduate or graduate students. A variety of innovations will also make the book of interest to specialists and researchers.

A lucid and concise account of the classification of algebraic surfaces, but expressed simply in the language of modern topology and sheaf theory. The ample number of exercises succeed both in giving the flavour of the extraordinary wealth of examples in the classical subject, and in equipping the reader with most of the techniques needed for research.

This book is essentially a self-contained 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. Parts of the book are suitable for a final year undergraduate course or for a master's level course.

This text is a self contained treatment of expander graphs and in particular their explicit construction. Expander graphs are both highly connected but sparse, and besides their interest within combinatorics and graph theory, they also find various applications in computer science and engineering.

This textbook is affordable and clearly illustrated, and is intended for advanced undergraduate or beginning graduate students with some previous experience of rings and fields. Alongside standard algebraic notions such as generators of modules and the ascending chain condition, the book develops in detail the geometric view of a commutative ring as the ring of functions on a space.

This 2003 book investigates the interplay between algebra and geometry is a beautiful area of mathematical investigation. Advances in computing and algorithms make it possible to tackle many classical problems in a down-to-earth and concrete fashion. This opens new vistas and allows us to pose, study and solve problems previously out of reach.

This book is an introduction to an important area of mathematics and mathematical physics. It is accessible to university students, but leads to topics of current research in the theory of harmonic maps. It is the first book on this subject at this level.

Fascinating connections exist between group theory and automata theory, and a wide variety of them are discussed in this text. Automata can be used in group theory to encode complexity, to represent aspects of underlying geometry on a space on which a group acts, and to provide efficient algorithms for practical computation. There are also many applications in geometric group theory. The authors provide background material in each of these related areas, as well as exploring the connections along a number of strands that lead to the forefront of current research in geometric group theory. Examples studied in detail include hyperbolic groups, Euclidean groups, braid groups, Coxeter groups, Artin groups, and automata groups such as the Grigorchuk group. This book will be a convenient reference point for established mathematicians who need to understand background material for applications, and can serve as a textbook for research students in (geometric) group theory.

Hurwitz theory, the study of analytic functions among Riemann surfaces, is a classical field and active research area in algebraic geometry. The subject's interplay between algebra, geometry, topology and analysis is a beautiful example of the interconnectedness of mathematics. This book introduces students to this increasingly important field, covering key topics such as manifolds, monodromy representations and the Hurwitz potential. Designed for undergraduate study, this classroom-tested text includes over 100 exercises to provide motivation for the reader. Also included are short essays by guest writers on how they use Hurwitz theory in their work, which ranges from string theory to non-Archimedean geometry. Whether used in a course or as a self-contained reference for graduate students, this book will provide an exciting glimpse at mathematics beyond the standard university classes.

The study of (special cases of) elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centres of research in number theory. This book, which is 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-Weil 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, as is the little that is needed on Galois cohomology. Many examples and exercises are included for the reader. For those new to elliptic curves, whether they are graduate students or specialists from other fields, this will be a fine introductory text.

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.

This book is an elementary introduction to the geometrical methods and notions used in special and general relativity. Particular emphasis is placed on the ideas concerned with the structure of spacetime and those which play a role in the Penrose-Hawking singularity theorems. Indeed, the author's prime purpose is to give a rigorous proof of the simplest of these theorems yet one which is representative of the whole. No previous exposure either to relativity theory or differential geometry is required to the reader, as necessary concepts are developed when needed, though in order to keep the presentation reasonably accessible, some restrictions are imposed on the types of space considered. Each chapter is liberally supplied with exercises and examples. The book will therefore be attractive to senior undergraduate and first year graduate students of mathematics and physics for independent study or courses preparatory to more advanced works.

Algebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. This short and readable introduction to algebraic geometry will be ideal for all undergraduate mathematicians coming to the subject for the first time. 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 non-singularity and dimension. He is at pains to stress the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book arises from an undergraduate course given at the University of Warwick and contains numerous examples and exercises illustrating the theory.

The area of nonlinear dispersive partial differential equations (PDEs) is a fast developing field which has become exceedingly technical in recent years. With this book, the authors provide a self-contained and accessible introduction for graduate or advanced undergraduate students in mathematics, engineering, and the physical sciences. Both classical and modern methods used in the field are described in detail, concentrating on the model cases that simplify the presentation without compromising the deep technical aspects of the theory, thus allowing students to learn the material in a short period of time. This book is appropriate both for self-study by students with a background in analysis, and for teaching a semester-long introductory graduate course in nonlinear dispersive PDEs. Copious exercises are included, and applications of the theory are also presented to connect dispersive PDEs with the more general areas of dynamical systems and mathematical physics.

Few books on the subject of Riemann surfaces cover the relatively modern theory of dessins d'enfants (children's drawings), which was launched by Grothendieck in the 1980s and is now an active field of research. In this 2011 book, the authors begin with an elementary account of the theory of compact Riemann surfaces viewed as algebraic curves and as quotients of the hyperbolic plane by the action of Fuchsian groups of finite type. They then use this knowledge to introduce the reader to the theory of dessins d'enfants and its connection with algebraic curves defined over number fields. A large number of worked examples are provided to aid understanding, so no experience beyond the undergraduate level is required. Readers without any previous knowledge of the field of dessins d'enfants are taken rapidly to the forefront of current research.

Celestial mechanics is the branch of mathematical astronomy devoted to studying the motions of celestial bodies subject to the Newtonian law of gravitation. This mathematical introductory textbook reveals that even the most basic question in celestial mechanics, the Kepler problem, leads to a cornucopia of geometric concepts: conformal and projective transformations, spherical and hyperbolic geometry, notions of curvature, and the topology of geodesic flows. For advanced undergraduate and beginning graduate students, this book explores the geometric concepts underlying celestial mechanics and is an ideal companion for introductory courses. The focus on the history of geometric ideas makes it perfect supplementary reading for students in elementary geometry and topology. Numerous exercises, historical notes and an extensive bibliography provide all the contextual information required to gain a solid grounding in celestial mechanics.