The Atiyah–Singer Index Theorem

The Abel Prize citation for Michael Atiyah and Isadore Singer reads: “The Atiyah–Singer index theorem is one of the great landmarks of twentieth-century mathematics, influencing profoundly many of the most important later developments in topology, differential geometry and quantum field theory”. This article is an attempt to describe the theorem, where it came from, its different manifestations and a collection of applications. It is clear from the citation that the theorem spans many areas. I have attempted to define in the text the most important concepts but inevitably a certain level of sophistication is needed to appreciate all of them. In the applications I have tried to indicate how one can use the theorem as a tool in a concrete fashion without necessarily retreating into the details of proof. This reflects my own appreciation of the theorem in its various forms as part of the user community. The vision and intuition that went into its proof is still a remarkable achievement and the Abel Prize is a true recognition of that fact.