There are few areas of pure mathematics which have profited more from the influx of new ideas from other disciplines than the subject of this book. The study of vector bundles over algebraic varieties has been stimulated by successive waves of migrant concepts over the last few years, largely coming from mathematical physics. It nevertheless also retains its roots in old questions concerning subvarieties of projective space, and this is a continuing source of challenging problems. The 1993 Durham Symposium Vector Bundles in Algebraic Geometry, sponsored by the LMS and SERC, had as its aim the goal of bringing together the leading researchers in the field to explore further these interactions: to see how old problems would yield to new techniques, and to present new opportunities for the already highly developed subject of algebraic geometry.

The present book is not, however, simply the Proceedings of that Symposium. Its purpose is certainly to reflect what was said and done there, but we hope that it also presents to the mathematical world an overview of the key areas of research involving vector bundles. Some of the principal speakers have been encouraged to expand their talks to give surveys of their respective areas. The reader can thus find here not only reports of recent progress, but also a perspective on where the new ideas have come from, what they are doing at the moment, and what they might be capable of in the future.