Atiyah-Singer index theorem. (Q2856948)

The aim of the paper is to explain the meaning of the Atiyah-Singer index theorem even to mathematicians not working in the field of differential geometry or topology. It is almost miraculous that the author was able to describe this theorem on 7 pages. The notion of differential operator and its symbol can be introduced on an open subset in a Euclidean space. For the notion of differentiable manifold the author refers the reader to another survey article in the same journal. Then he devotes a lot of effort to the notions of vector bundle over a differentiable manifold and to differential operators among them. Differential operators between two vector bundles are introduced as ordinary differential operators via trivializations of vector bundles. Fortunately, with the analytical index there is no great problem. It is much more difficult to define the topological index. There was no other possibility than only to sketch the relevant construction. Then it is already possible to formulate the index theorem. In the end we find three examples showing at least partially the wide field of applications of the index theorem.