These notes are based on the course of lectures I gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. In fact, rational cohomology is defined in terms of K-theory.

These Lecture Notes are an expanded version of the Fermi Lectures Atiyah gave at Scuola Normale Superiore in Pisa, the Loeb Lectures at Harvard and the Whittemore Lectures at Yale, in 1978. In all cases he was addressing a mixed audience of mathematicians and physicists and the presentation had to be tailored accordingly. Throughout, Atiyah presented the mathematical material in a somewhat unorthodox order, following a pattern which he felt would relate the new techniques to familiar ground for physicists.
The main new results presented in the lectures, namely the construction of all multi-istanton solutions of Yang-Mills fields, is the culmination of several years of fruitful interaction between many physicists and mathematicians. The major breakthrough came with the observation by Ward that the complex methods developed by Penrose in his “twistor programme” were ideally suited to the study of the Yang-Mills equations. The instanton problem was then seen to be equivalent to a problem in complex analysis and to one in algebraic geometry. Using the powerful methods of modern algebraic geometry it was not long before the problem was finally solved.

Paul Adrien Maurice Dirac was one of the founders of quantum theory. He is numbered alongside Newton, Maxwell and Einstein as one of the greatest physicists of all time. Together the lectures in this volume, originally presented on the occasion of the dedication ceremony for a plaque commemorating Dirac in Westminster Abbey, give a unique insight into the relationship between Dirac's character and his scientific achievements. The text begins with the dedication address given by Stephen Hawking at the ceremony. Then Abraham Pais describes Dirac as a person and his approach to his work. Maurice Jacob explains how Dirac was led to introduce the concept of antimatter, and its central role in modern particle physics and cosmology, followed by an account by David Olive of the origin and enduring influence of Dirac's work on magnetic monopoles. Finally, Sir Michael Atiyah explains the deep and widespread significance of the Dirac equation in mathematics.

Professor Atiyah is one of the greatest living mathematicians and is well known throughout the mathematical world. He is a recipient of the Fields Medal, the mathematical equivalent of the Nobel Prize, and is still at the peak of his career. His huge number of published papers, focusing on the areas of algebraic geometry and topology, have here been collected into six volumes, divided thematically for easy reference by individuals interested in a particular subject. This six volume set of the collected works of Professor Sir Michael Atiyah, includes: "Collected Works - Volume 1 - Early Papers, General Papers"; "Collected Works - Volume 2 - K-Theory"; "Collected Works - Volume 3 - Index Theory - 1"; "Collected Works - Volume 4 - Index Theory - 2"; and "Collected Works - Volume 5 - Gauge Theories". "Collected Works: Volume 6" includes publications since 1987, including his work on skyrmions, "Atiyah's axioms" for topological quantum field theories, monopoles, knots, K-theory, equivariant problems, point particles, and M-theory.

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.