Bücher der Reihe London Mathematical Society Lecture Note Series

Singularity theory is not a field in itself, but rather an application of algebraic geometry, analytic geometry and differential analysis. The adjective 'singular' in the title refers here to singular points of complex-analytic or algebraic varieties or mappings. A tractable (and very natural) class of singularities to study are the isolated complete intersection singularities, and much progress has been made over the past decade in understanding these and their deformations.

The interaction between ergodic theory and discrete groups has a long history and much work was done in this area by Hedlund, Hopf and Myrberg in the 1930s. There has been a great resurgence of interest in the field, due in large measure to the pioneering work of Dennis Sullivan. Tools have been developed and applied with outstanding success to many deep problems. The ergodic theory of discrete groups has become a substantial field of mathematical research in its own right, and it is the aim of this book to provide a rigorous introduction from first principles to some of the major aspects of the theory. The particular focus of the book is on the remarkable measure supported on the limit set of a discrete group that was first developed by S. J. Patterson for Fuchsian groups, and later extended and refined by Sullivan.

Papers from the Symposium on stochastic analysis, which took place at the University of Durham in July 1990.

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 1977 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. This volume contains much to interest mathematicians and theoretical physicists from advanced undergraduate level upwards.

This book considers the theory of 'integrable' non-linear partial differential equations. The theory was developed at first by mathematical physicists but later mathematicians, particularly from the Soviet Union, were attracted to the field. In this volume are reprinted some fundamental contributions, originally published in Russian Mathematical Surveys, from some of the leading Soviet workers. Dr George Wilson has written an introduction intended to smooth the reader's path through some of the articles.

The unifying theme of this collection of papers by the very creative Russian mathematician I. M. Gelfand and his co-workers is the representation theory of groups and lattices. Two of the papers were inspired by application to theoretical physics; the others are pure mathematics though all the papers will interest mathematicians at quite opposite ends of the subject.

Using a largely homological approach, this book focuses on the representation theory of q-Schur algebras and connections with the representation theory of Hecke algebras and quantum general linear groups. It will be primarily of interest to researchers in algebra and related topics in pure mathematics.

Highlighting important progress being made in areas such as the local Langlands programme and automorphic forms, this timely volume is indispensable for researchers in the area of L-functions and Galois representations. Articles include Khare on Serre's conjecture, Yafaev on the Andre-Oort conjecture, and Emerton on Jacquet functors.

Vector bundles and their associated moduli spaces are of fundamental importance in algebraic geometry. This volume, authored by some of the subject's leading experts, covers foundational material as well as focusing on research topics currently at the forefront of the field. Suitable for both graduate students and more experienced researchers.

Leading figures in modern computational mathematics present here the latest research and give surveys of its contemporary topics. This is a valuable resource for all working in numerical analysis, optimization, computer algebra and scientific computing.

Intended for researchers and graduate students in theoretical computer science and mathematical logic, this volume contains accessible surveys by leading researchers from areas of current work in logical aspects of computer science, where both finite and infinite model-theoretic methods play an important role. The chapters include extensive bibliographies.

An introduction to the behaviour of random matrices. Suitable for postgraduate students and non-experts.

This is a one-stop introduction to the methods of ergodic theory applied to holomorphic iteration. The authors begin with introductory chapters presenting the necessary tools from ergodic theory thermodynamical formalism, and then focus on recent developments in the field of 1-dimensional holomorphic iterations and underlying fractal sets, from the point of view of geometric measure theory and rigidity. Detailed proofs are included. Developed from university courses taught by the authors, this book is ideal for graduate students. Researchers will also find it a valuable source of reference to a large and rapidly expanding field. It eases the reader into the subject and provides a vital springboard for those beginning their own research. Many helpful exercises are also included to aid understanding of the material presented and the authors provide links to further reading and related areas of research.

Survey and research articles from the 1999 Bielefeld conference on topological, combinatorial and arithmetic aspects of (infinite) groups. All contributions are written in a relaxed and attractive style, accessible to specialists and also post-graduate students, who will find inspiration for a number of research projects at various levels of difficulty.

This volume provides an up-to-date overview of current research in several areas of combinatorics, ranging from combinatorial number theory to geometry. The authors are some of the world's foremost researchers in their fields, and here they summarize existing results, and give a unique preview of work currently being written up.

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

This technique is used in the solution of boundary problems for partial differential equations. Its applications include the Dirac theory of quantum mechanics. The author discusses connections to the theory of C*-algebras and the relation of the hyperbolic theory to the propagation of maximal ideals.

These volumes contain selected papers presented at the international conference on group theory held in St Andrews in 1989. The themes of the conference were combinatorial and computational group theory; four leading group theorists (J. A. Green, N. D. Gupta, O. H. Kegel and J. G. Thompson) gave courses whose content is reproduced here.

This volume contains the invited papers presented at the British Combinatorial Conference, held at the University of Surrey in July 1991. The papers will provide excellent reading for all those interested in combinatorics.

Boolean function complexity has seen exciting advances in the last few years. It is a long established area of discrete mathematics which uses combinatorial and occasionally algebraic methods.

A collection of expository surveys given by some distinguished mathematicians during the 1990 International Mathematical Congress.

In this book Professor Kempf gives an introduction to the theory of algebraic functions on varieties from a sheaf theoretic standpoint. By taking this view he is able to give a clean and lucid account of the subject which will be easily accessible to all newcomers to algebraic varieties, graduate students or experts from other fields alike. Anyone who goes on to study schemes will find that this book is an ideal preparatory text.

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

This is amongst the first books on the theory of prehomogeneous vector spaces, and represents the author's deep study of the subject.

This is a collection of thirty-five articles on covering topics such as finite projective spaces, generalized polygons, strongly regular graphs, diagram geometries, and polar spaces. Included here are articles from many of the leading practitioners in the field including, for the first time, several distinguished Russian mathematicians.

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

Stochastic partial differential equations can be used in many areas of science to model complex systems that evolve over time. Their analysis is currently an area of much research interest. This book consists of papers given at the ICMS meeting held in 1994 on this topic and it brings together some of the world's best known authorities on stochastic partial differential equations.

These articles give an up-to-date overview of combinatorics that will be extremely useful to both mathematicians and computer scientists.

The 1993 Durham Symposium on Vector Bundles in Algebraic Geometry brought together some of the leading researchers in the field to explore further these interactions. This book is a collection of survey articles by the main speakers at the symposium and presents to the mathematical world an overview of the key areas of research involving vector bundles.

These proceedings give a state-of-the-art account of the area of finite fields and their applications in communications (coding theory, cryptology), combinatorics, design theory, quasirandom points, algorithms and their complexity. This volume is an invaluable resource for any researcher in finite fields or related areas.