Index theorem. 1

The Atiyah-Singer index theorem is a remarkable result that allows one to compute the space of solutions of a linear elliptic partial differential operator on a manifold in terms of purely topological data related to the manifold and the symbol of the operator. First proved by Atiyah and Singer in 1963, it marked the beginning of a completely new direction of research in mathematics with relations to differential geometry, partial differential equations, differential topology, K-theory, physics, and other areas. The author's main goal in this volume is to give a complete proof of the index theorem. The version of the proof he chooses to present is the one based on the localization theorem. The prerequisites include a first course in differential geometry, some linear algebra, and some facts about partial differential equations in Euclidean spaces.