Geometry and Spectra of Compact Riemann Surfaces (Modern Birkhäuser Classics)

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This book deals with two subjects. The first subject is the geometric theory of compact Riemann surfaces of genus greater than one, the second subject is the Laplace operator and its relationship with the geometry of compact Riemann surfaces. The book grew out of the idea, a long time ago, to publish a Habili- tionsschrift, a thesis, in which I studied Bers' pants decomposition theorem and its applications to the spectrum of a compact Riemann surface.

As this approach to the geometry of a compact Riemann surface did not exist in book form, I took this book as an occasion to carry out the geometry in detail, and so it grew by several chapters. In this book, research mathematicians in complex geometry, hyperbolic geometry and Teichmuller spaces will find a selection of strong papers by international experts. Ablowitz, P. Clarkson This book brings together several aspects of soliton theory currently available only in research papers.

Emphasis is given to the multi-dimensional problems which arise and includes inverse scattering in multi-dimensions, integrable nonlinear evolution equations in multi-dimensions and the dbar method. Ablowitz, Harvey Segur A study of the inverse scattering transform and its application to problems of nonlinear dispersive waves that arise in fluid dynamics, plasma physics, nonlinear optics, particle physics, crystal lattice theory, nonlinear circuit theory and other areas. Accola The book's main concern is automorphisms of Riemann surfaces, giving a foundational treatment from the point of view of Galois coverings, and treating the problem of the largest automorphism group for a Riemann surface of a given genus.

In addition, the extent to which fixed points of automorphisms are generalized Weierstrass points is considered. The extremely useful inequality of Castelnuovo- Severi is also treated. While the methods are elementary, much of the material does not appear in the current texts on Riemann surfaces, algebraic curves. The book is accessible to a reader who has had an introductory course on the theory of Riemann surfaces or algebraic curves.

Moduli of Abelian Varieties Allan Adler, Sundararaman Ramanan This is a book aimed at researchers and advanced graduate students in algebraic geometry, interested in learning about a promising direction of research in algebraic geometry. It begins with a generalization of parts of Mumford's theory of the equations defining abelian varieties and moduli spaces.

It shows through striking examples how one can use these apparently intractable systems of equations to obtain satisfying insights into the geometry and arithmetic of these varieties. It also introduces the reader to some aspects of the research of the first author into representation theory and invariant theory and their applications to these geometrical questions.

Agarwal, Claudio Cuevas, Carlos Lizama This work introduces readers to the topic of maximal regularity for difference equations.

The authors systematically present the method of maximal regularity, outlining basic linear difference equations along with relevant results. They address recent advances in the field, as well as basic semi group and cosine operator theories in the discrete setting. The authors also identify some open problems that readers may wish to take up for further research. This book is intended for graduate students and researchers in the area of difference equations, particularly those with advance knowledge of and interest in functional analysis.

Complex Analysis Lars Ahlfors A standard source of information of functions of one complex variable, this text has retained its wide popularity in this field by being consistently rigorous without becoming needlessly concerned with advanced or overspecialized material. Difficult points have been clarified, the book has been reviewed for accuracy, and notations and terminology have been modernized. Chapter 2, Complex Functions, features a brief section on the change of length and area under conformal mapping, and much of Chapter 8, Global-Analytic Functions, has been rewritten in order to introduce readers to the terminology of germs and sheaves while still emphasizing that classical concepts are the backbone of the theory.

Chapter 4, Complex Integration, now includes a new and simpler proof of the general form of Cauchy's theorem. There is a short section on the Riemann zeta function, showing the use of residues in a more exciting situation than in the computation of definite integrals. Advances in the Theory of Riemann Surfaces. Lars Valerian Ahlfors, Lipman Bers Intended for researchers in Riemann surfaces, this volume summarizes a significant portion of the work done in the field during the years to Studies in Modern Algebra A.

Alperin Representation theory has applications to number theory, combinatorics and many areas of algebra. The aim of this text is to present some of the key results in the representation theory of finite groups. Professor Alperin concentrates on local representation theory, emphasizing module theory throughout. In this way many deep results can be obtained rather quickly.

After two introductory chapters, the basic results of Green are proved, which in turn lead in due course to Brauer's First Main Theorem. A proof of the module form of Brauer's Second Main Theorem is then presented, followed by a discussion of Feit's work connecting maps and the Green correspondence. The work concludes with a treatment, new in part, of the Brauer-Dade theory. Exercises are provided at the end of most sections; the results of some are used later in the text.

Products of Groups Bernhard Amberg, Silvana Franciosi, Francesco de Giovanni Groups comprising two subcomponents are of particular interest to group theorists who want to know in what way the structure of the product is related to that of its subgroups. This monograph gives the first detailed account of the most important results of group product research from the past 35 years.

Although the emphasis is on infinite groups, relevant theorems about finite products of groups are also proved. This book will be of interest to research students and specialists in group theory, and will be useful in seminars or as a supplement in courses in general group theory.

A special chapter on conjugacy and splitting theorems obtained by means of the cohomology of groups has never before appeared in book form. Number Theory George E. Andrews Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic. In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to elementary number theory.

In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent simplicity of the proofs for many theorems. Among the topics covered in this accessible, carefully designed introduction are multiplicativity-divisibility, including the fundamental theorem of arithmetic, combinatorial and computational number theory, congruences, arithmetic functions, primitive roots and prime numbers. Later chapters offer lucid treatments of quadratic congruences, additivity including partition theory and geometric number theory.

Of particular importance in this text is the author's emphasis on the value of numerical examples in number theory and the role of computers in obtaining such examples.

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Exercises provide opportunities for constructing numerical tables with or without a computer. Students can then derive conjectures from such numerical tables, after which relevant theorems will seem natural and well-motivated.. Kleinian groups are, classically, discrete groups of conformal automorphisms of the Riemann sphere, and these can be regarded too as being groups of holomorphic automorphisms of the complex projective line CP1.

When going into higher dimensions, there is a dichotomy: Should we look at conformal automorphisms of the n-sphere? These two theories are different in higher dimensions. In the first case we are talking about groups of isometries of real hyperbolic spaces, an area of mathematics with a long-standing tradition. In the second case we are talking about an area of mathematics that still is in its childhood, and this is the focus of study in this monograph. This brings together several important areas of mathematics, as for instance classical Kleinian group actions, complex hyperbolic geometry, chrystallographic groups and the uniformization problem for complex manifolds.

Apostol "This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory.


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For this reason, the book starts with the most elementary properties of the natural integers. Nevertheless, the text succeeds in presenting an enormous amount of material in little more than pages. Apostol A new edition of a classical treatment of elliptic and modular functions with some of their number-theoretic applications, this text offers an updated bibliography and an alternative treatment of the transformation formula for the Dedekind eta function.

The articles collected here feature recent developments in various areas of non-Archimedean analysis: Hilbert and Banach spaces, finite dimensional spaces, topological vector spaces and operator theory, strict topologies, spaces of continuous functions and of strictly differentiable functions, isomorphisms between Banach functions spaces, and measure and integration.

Many long-standing problems have been solved using the general techniques developed in algebraic geometry during the 's and 's. Additionally, unexpected and deep connections between algebraic curves and differential equations have been uncovered, and these in turn shed light on other classical problems in curve theory.

These books give a presentation of one of the central areas of this recent activity; namely, the study of linear series on both a fixed curve Volume I and on a variable curve Volume II. Our goal is to give a comprehensive and self-contained account of the extrinsic geometry of algebraic curves, which in our opinion constitutes the main geometric core of the recent advances in curve theory. Basic Topology M. Armstrong In this broad introduction to topology, the author searches for topological invariants of spaces, together with techniques for their calculating.

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Students with knowledge of real analysis, elementary group theory, and linear algebra will quickly become familiar with a wide variety of techniques and applications involving point-set, geometric, and algebraic topology. Over illustrations and more than problems of various difficulties help students gain a thorough understanding of the subject. Mathematical Methods of Classical Mechanics V. Arnold This book constructs the mathematical apparatus of classical mechanics from the beginning, examining basic problems in dynamics like the theory of oscillations and the Hamiltonian formalism.

The author emphasizes geometrical considerations and includes phase spaces and flows, vector fields, and Lie groups. Discussion includes qualitative methods of the theory of dynamical systems and of asymptotic methods like averaging and adiabatic invariance. Geometric Algebra E. Artin This classic text, written by one of the foremost mathematicians of the 20th century, is now available in a low-priced paperback edition.

Exposition is centered on the foundations of affine geometry, the geometry of quadratic forms, and the structure of the general linear group. Context is broadened by the inclusion of projective and symplectic geometry and the structure of symplectic and orthogonal groups.

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Milgram In the nineteenth century, French mathematician Evariste Galois developed the Galois theory of groups-one of the most penetrating concepts in modem mathematics. The elements of the theory are clearly presented in this second, revised edition of a volume of lectures delivered by noted mathematician Emil Artin. The book has been edited by Dr. Arthur N. Milgram, who has also supplemented the work with a Section on Applications.

The first section deals with linear algebra, including fields, vector spaces, homogeneous linear equations, determinants, and other topics. A second section considers extension fields, polynomials, algebraic elements, splitting fields, group characters, normal extensions, roots of unity, Noether equations, Jummer's fields, and more.

Milgram's section on applications discusses solvable groups, permutation groups, solution of equations by radicals, and other concepts. A Short Course on Spectral Theory William Arveson This book presents the basic tools of modern analysis within the context of the fundamental problem of operator theory: to calculate spectra of specific operators on infinite dimensional spaces, especially operators on Hilbert spaces. Products of Finite Groups Mohamed Asaad The study of finite groups factorised as a product of two or more subgroups has become a subject of great interest during the last years with applications not only in group theory, but also in other areas like cryptography and coding theory.

It has experienced a big impulse with the introduction of some permutability conditions. The aim of this book is to gather, order, and examine part of this material, including the latest advances made, give some new approach to some topics, and present some new subjects of research in the theory of finite factorised groups. Representation Theory of Lie Groups M. Atiyah, R. Bott, S. Helgason, D. Kazhdan, B. Kostant, G. Lustztig Lie groups and their representations occupy an important place in mathematics with applications in such diverse fields as differential geometry, number theory, differential equations and physics.

In a symposium was held in Oxford to introduce this rapidly developing and expanding subject to non-specialists. This volume contains the lectures of ten distinguished mathematicians designed to provide the reader with a deeper understanding of the fundamental theory and appreciate the range of results.


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This volume contains much to interest mathematicians and theoretical physicists from advanced undergraduate level upwards. Introduction To Commutative Algebra Michael Atiyah This book grew out of a course of lectures given to third year undergraduates at Oxford University and it has the modest aim of producing a rapid introduction to the subject. It is designed to be read by students who have had a first elementary course in general algebra.

On the other hand, it is not intended as a substitute for the more voluminous tracts such as Zariski-Samuel or Bourbaki. We have concentrated on certain central topics, and large areas, such as field theory, are not touched. In content we cover rather more ground than Northcott and our treatment is substantially different in that, following the modern trend, we put more emphasis on modules and localization. Holomorphic Curves in Symplectic Geometry Michele Audin, Jacques Lafontaine This book is devoted to pseudo-holomorphic curve methods in symplectic geometry.

It contains an introduction to symplectic geometry and relevant techniques of Riemannian geometry, proofs of Gromov's compactness theorem, an investigation of local properties of holomorphic curves, including positivity of intersections, and applications to Lagrangian embeddings problems. The chapters are based on a series of lectures given previously by the authors M. Audin, A. Banyaga, P. Gauduchon, F.

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Labourie, J. Lafontaine, F. Lalonde, Gang Liu, D. McDuff, M. Muller, P. Pansu, L. Polterovich, J. In an attempt to make this book accessible also to graduate students, the authors provide the necessary examples and techniques needed to understand the applications of the theory.

Books by Peter Buser

The exposition is essentially self-contained and includes numerous exercises. Symplectic manifolds and torus actions are investigated, with numerous examples of torus actions, for instance on some moduli spaces. Although the book is still centered on convexity results, it contains much more material, in particular lots of new examples and exercises. Linear Algebra Done Right Sheldon Axler This text for a second course in linear algebra, aimed at math majors and graduates, adopts a novel approach by banishing determinants to the end of the book and focusing on understanding the structure of linear operators on vector spaces.

The author has taken unusual care to motivate concepts and to simplify proofs. For example, the book presents - without having defined determinants - a clean proof that every linear operator on a finite-dimensional complex vector space has an eigenvalue. The book starts by discussing vector spaces, linear independence, span, basics, and dimension. Students are introduced to inner-product spaces in the first half of the book and shortly thereafter to the finite- dimensional spectral theorem. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra.

This second edition features new chapters on diagonal matrices, on linear functionals and adjoints, and on the spectral theorem; some sections, such as those on self-adjoint and normal operators, have been entirely rewritten; and hundreds of minor improvements have been made throughout the text. Wilson Introduction to Classical Integrable Systems Olivier Babelon, Denis Bernard, Michel Talon Introducing the reader to classical integrable systems and their applications, this book synthesizes the different approaches to the subject, providing a set of interconnected methods for solving problems in mathematical physics.

The authors introduce and explain each method, and demonstrate how it can be applied to particular examples. Rather than presenting an exhaustive list of the various integrable systems, they focus on classical objects which have well-known quantum counterparts, or are the semi-classical limits of quantum objects. They thus enable readers to understand the literature on quantum integrable systems. Bailey, Anthony W. Knapp This book is a course in representation theory of semisimple groups, automorphic forms and the relations between these two subjects written by some of the world's leading experts in these fields.

It is based on the instructional conference of the International Centre for Mathematical Sciences in Edinburgh. The book begins with an introductory treatment of structure theory and ends with an essay by Robert Langlands on the current status of functoriality. All papers are intended to provide overviews of the topics they address, and the authors have supplied extensive bibliographies to guide the reader who wants more detail. The aim of the articles is to treat representation theory with two goals in mind: to help analysts make systematic use of Lie groups in work on harmonic analysis, differential equations, and mathematical physics and to provide number theorists with the representation-theoretic input to Wiles' proof of Fermat's Last Theorem.

This book features discussion of representation theory from many experts' viewpoints; treatment of the subject from the foundations through recent advances; discussion of the analogies between analysis of cusp forms and analysis on semisimple symmetric spaces, which have been at the heart of research breakthroughs for 40 years; and, extensive bibliographies.

In our century the methods and ideas of topology, commutative algebra and Grothendieck's schemes enriched it and seemed to have replaced once and forever the somewhat naive language of classical algebraic geometry. This classic book, written in , covers the whole of algebraic geometry and associated theories. Baker discusses the subject in terms of transcendental functions, and theta functions in particular. Many of the ideas put forward are of continuing relevance today, and some of the most exciting ideas from theoretical physics draw on work presented here. Abelian Varieties Wolf P.

Bartle Presents the basic theory of real analysis. The algebraic and order properties of the real number system are presented in a simpler fashion than in the previous edition. Topics in Combinatorial Group Theory Gilbert Baumslag Combinatorial group theory is a loosely defined subject, with close connections to topology and logic. With surprising frequency, problems in a wide variety of disciplines, including differential equations, automorphic functions and geometry, have been distilled into explicit questions about groups, typically of the following kind: Are the groups in a given class finite e.

Finitely generated? Finitely presented? What are the conjugates of a given element in a given group? What are the subgroups of that group? Is there an algorithm for deciding for every pair of groups in a given class whether they are isomorphic or not? The objective of combinatorial group theory is the systematic development of algebraic techniques to settle such questions.

In view of the scope of the subject and the extraordinary variety of groups involved, it is not surprising that no really general theory exists. These notes, bridging the very beginning of the theory to new results and developments, are devoted to a number of topics in combinatorial group theory and serve as an introduction to the subject on the graduate level. Beardon This text is intended to serve as an introduction to the geometry of the action of discrete groups of Mobius transformations. The subject matter has now been studied with changing points of emphasis for over a hundred years, the most recent developments being connected with the theory of 3-manifolds: see, for example, the papers of Poincare [77] and Thurston [].

About , the now well-known but virtually unobtainable Fenchel-Nielsen manuscript appeared. Further, wherever possible, results should be stated in a form that is invariant under conjugation, thus making the intrinsic nature of the result more apparent. Despite the fact that the subject matter is concerned with groups of isometries of hyperbolic geometry, many publications rely on Euclidean estimates and geometry.

However, the recent developments have again emphasized the need for hyperbolic geometry, and I have included a comprehensive chapter on analytical not axiomatic hyperbolic geometry. It is hoped that this chapter will serve as a "dictionary" offormulae in plane hyperbolic geometry and as such will be of interest and use in its own right. Thompson published a proof of a conjecture by Burnside that every finite group of odd order is solvable.

This proof, which ran for pages, was a tour-de-force of mathematics and inspired intense effort to classify finite simple groups. This book presents a revision and expansion of the first half of the proof of the Feit-Thompson theorem. Simpler, more detailed proofs are provided for some intermediate theorems. Recent results are used to shorten other proofs. Finite Permutation Groups Trans. This volume also contains a kaleidoscope of papers based on several of the more important talks presented at the meeting.

It provides a cutting-edge connection to some of the most important recent developments in the field. Through a combination of survey papers, research articles, and extensive references to earlier work, this volume allows the reader to quickly gain an overview of current activity in the field and become acquainted with many of the recent sub-branches of its development. Complex Abelian Varieties Christina Birkenhake, Herbert Lange This book explores the theory of abelian varieties over the field of complex numbers, explaining both classic and recent results in modern language.

The second edition adds five chapters on recent results including automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture. Bloch This book is the long-awaited publication of the famous Irvine lectures. Delivered in at the University of California at Irvine, these lectures turned out to be an entry point to several intimately-connected new branches of arithmetic algebraic geometry, such as regulators and special values of L-functions of algebraic varieties, explicit formulas for them in terms of polylogarithms, the theory of algebraic cycles, and eventually the general theory of mixed motives which unifies and underlies all of the above and much more.

In the 20 years since, the importance of Bloch's lectures has not diminished. A lucky group of people working in the above areas had the good fortune to possess a copy of old typewritten notes of these lectures. Now everyone can have their own copy of this classic work. Bobenko, Christian Klein This volume offers a well-structured overview of existent computational approaches to Riemann surfaces and those currently in development.

The authors of the contributions represent the groups providing publically available numerical codes in this field. Thus this volume illustrates which software tools are available and how they can be used in practice. In addition examples for solutions to partial differential equations and in surface theory are presented. The intended audience of this book is twofold. It can be used as a textbook for a graduate course in numerics of Riemann surfaces, in which case the standard undergraduate background, i.

In particular, no knowledge of the theory of Riemann surfaces is expected; the necessary background in this theory is contained in the Introduction chapter. At the same time, this book is also intended for specialists in geometry and mathematical physics applying the theory of Riemann surfaces in their research.

It is the first book on numerics of Riemann surfaces that reflects the progress made in this field during the last decade, and it contains original results. There are a growing number of applications that involve the evaluation of concrete characteristics of models analytically described in terms of Riemann surfaces.

Many problem settings and computations in this volume are motivated by such concrete applications in geometry and mathematical physics. Casselman Contains sections on Reductive groups, representations, Automorphic forms and representations. Algebraic Geometry and Commutative Algebra Siegfried Bosch Algebraic geometry is a fascinating branch of mathematics that combines methods from both, algebra and geometry. It transcends the limited scope of pure algebra by means of geometric construction principles. The scheme-theoretic approach to algebraic geometry is explained for non-experts.

More advanced readers can use the book to broaden their view on the subject. A separate part deals with the necessary prerequisites from commutative algebra. On a whole, the book provides a very accessible and self-contained introduction to algebraic geometry, up to a quite advanced level. Every chapter of the book is preceded by a motivating introduction with an informal discussion of the contents. Typical examples and an abundance of exercises illustrate each section.

This way the book is an excellent solution for learning by yourself or for complementing knowledge that is already present. It can equally be used as a convenient source for courses and seminars or as supplemental literature. Bourbaki From the reviews of the French edition: "This is a rich and useful volume. Seligman in MathReviews. Bredon This book offers an introductory course in algebraic topology.

Starting with general topology, it discusses differentiable manifolds, cohomology, products and duality, the fundamental group, homology theory, and homotopy theory. From the reviews: "An interesting and original graduate text in topology and geometry A beginning graduate student can use this text to learn a great deal of mathematics. He provides classifications of all automorphism groups up to genus The book also classifies the ordinary characters for several groups, arising from the action of automorphisms on the space of holomorphic abelian differentials of a compact Reimann surface.

This book is suitable for graduate students and researchers in group theory, representation theory, complex analysis and computer algebra. Characters and Automorphism Groups of Compact Riemann Surfaces Thomas Breuer The author uses modern methods from computational group theory and representation theory to treat this classical topic of function theory. Families of Automorphic Forms Roelof W. Bruggeman Automorphic forms on the upper half plane have been studied for a long time. Most attention has gone to the holomorphic automorphic forms, with numerous applications to number theory.

Maass, [34], started a systematic study of real analytic automorphic forms. The names Selberg and Roelcke are connected to the spectral theory of real analytic automorphic forms, see, e. This culminates in the trace formula of Selberg, see, e.

Automorphicformsarefunctionsontheupperhalfplanewithaspecialtra- formation behavior under a discontinuous group of non-euclidean motions in the upper half plane. One may ask how automorphic forms change if one perturbs this group of motions. This question is discussed by, e.

Hejhal also discusses the e? In [5]—[7] I considered variation of automorphic forms for the full modular group under perturbation of the m- tiplier system. The present study arose from a plan to extend these results to much more general groups discrete co? Theta Functions Bowdoin Me. Theta Functions, Bowdoin Me. Gunning, Leon Ehrenpreis During his long and productive career, Salomon Bochner worked in a variety of different areas of mathematics.

This four part set brings together his collected papers, illustrating the range and depth of his mathematical interests. The books are available either individually or as a set. Buchstaber, S. Novikov This collection contains articles reflecting the most recent activity in topology and mathematical physics presented at the S.

Novikov Seminar held in Moscow. Papers in the volume are devoted to problems in geometry, topology, and mathematical physics, including applications of topology to physical problems. Such a combination is a long-standing tradition of the seminar, which originated in Bujalance, A. Costa, E. It provides a mixture of classical material, recent results and some non-mainstream topics.

Etayo, Jose M. Gamboa, Grzegorz Gromadzki This research monograph provides a self-contained approach to the problem of determining the conditions under which a compact bordered Klein surface S and a finite group G exist, such that G acts as a group of automorphisms in S. The cases dealt with here take G cyclic, abelian, nilpotent or supersoluble and S hyperelliptic or with connected boundary.

No advanced knowledge of group theory or hyperbolic geometry is required and three introductory chapters provide as much background as necessary on non-euclidean crystallographic groups. The graduate reader thus finds here an easy access to current research in this area as well as several new results obtained by means of the same unified approach.

It examines the number of conjugacy classes of symmetries, the numbers of ovals of symmetries and the symmetry types of Riemann surfaces. Lie Groups Daniel Bump This book proceeds beyond the representation theory of compact Lie groups which is the basis of many texts and offers a carefully chosen range of material designed to give readers the bigger picture.

It explores compact Lie groups through a number of proofs and culminates in a "topics" section that takes the Frobenius-Schur duality between the representation theory of the symmetric group and the unitary groups as unifying them. Burkill, H. Burkill The classic analysis textbook from Burkill and Burkill is now available in the Cambridge Mathematical Library. This straightforward course, based on the idea of a limit, is for students of mathematics and physics who have acquired a working knowledge of calculus and are ready for a more systematic approach.

The treatment given here also brings in other limiting processes, such as the summation of infinite series and the expansion of trigonometric functions as power series. Particular attention is given to clarity of exposition and the logical development of the subject matter. This publication is professionally scanned from an original edition of the book, and of the best possible quality.

This popular classic work by William Burnside is in the English language. If you enjoy the works of William Burnside then we highly recommend this publication for your book collection. Elementary Number Theory David Burton Elementary Number Theory, Seventh Edition, is written for the one-semester undergraduate number theory course taken by math majors, secondary education majors, and computer science students.

This contemporary text provides a simple account of classical number theory, set against a historical background that shows the subject's evolution from antiquity to recent research. An introduction to abstract mathematical systems David M. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces.

Research workers and graduate students interested in compact Riemann surfaces will find here a number of useful tools and insights to apply to their investigations. This volume treats linear representations of finite reductive groups and their modular aspects together with Hecke algebras, complex reflection groups, quantum groups, arithmetic groups, Lie groups, symmetric groups and general finite groups.

Campbell, E. Robertson, N. Ruskuc, G. Smith This two-volume book contains selected papers from the international conference "Groups St. Andrews in Bath". The articles are arranged in roughly alphabetical order and cover a wide spectrum of modern group theory. There are articles based on lecture courses given by five main speakers together with refereed survey and research articles contributed by other conference participants.

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Proceedings of earlier "Groups St. Andrews" conferences have had a major impact on the development of group theory and these volumes should be equally important.

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Campbell, M. Quick, E. Robertson, C. Roney-Dougal, G. Smith, G. Traustason Groups St Andrews was held in the University of Bath in August and this first volume of a two-volume book contains selected papers from the international conference. Five main lecture courses were given at the conference, and articles based on their lectures form a substantial part of the proceedings.

Apart from the main speakers, refereed survey and research articles were contributed by other conference participants. Arranged in alphabetical order, these articles cover a wide spectrum of modern group theory. The regular proceedings of Groups St Andrews conferences have provided snapshots of the state of research in group theory throughout the past 30 years.