in the late 1920's the relentless march ofideas and discoveries had carried physics to a generally acceptedrelativistic theory of the electron. the physicist p.a.m. dirac,however, was dissatisfied with the prevailing ideas and, somewhatin isolation, sought for a better formulation. by 1928 he succeededin finding a theory which accorded with his own ideas and also fitmost of the established principles of the time. ultimately thistheory proved to be one of the great intellectual achievements ofthe period. it was particularly remarkable for the internal beautyof its mathematical structure which not only clarified muchpreviously mysterious phenomena but also predicted in a compellingway the existence of an electron-like particle of negative energy.indeed such particles were subsequently found to exist and ourunderstanding of nature was transformed.
　　 because of its compelling beauty and physical significance it isperhaps not surprising that the ideas at the heart of dirac'stheory have also been discovered to play a role of great importancein modern mathematics, particularly in the interrelations betweentopology, geometry and analysis. a great part of this newunderstanding comes from the work of m. atiyah and i. singer. it istheir work and its implications which form the focus of thisbook.