Intuitive

Concrete

Abstract

Why is it interesting?

The Atiyah-Singer index theorem tells one the number of solutions to a large class of differential equations purely in terms of topology. Topology is that part of mathematics that deals with those aspects of geometrical objects that don't change as one deforms the object (the standard explanatory joke is that 'a topologist is someone who can't tell the difference between a coffee cup and a doughnut'). One important aspect of the index theorem is that it can be proved by relating the differential equation under study to a generalised version of the Dirac equation. Atiyah and Singer rediscovered the Dirac equation for themselves during their work on the theorem. Their theorem says that one can calculate the number of solutions of an equation by finding the number of solutions of the related generalised Dirac equation. It was for these generalised Dirac equations that they found a beautiful topological formula for the number of their solutions. […] Atiyah became interested [in Yang-Mills theory], and they quickly realised that their index theorem could be applied in this case and it allowed a determination of exactly how many solutions the self-duality equations would have.

page 118 in Not Even Wrong by Peter Woit

The next major development in the story of the so-far-only-physicist’s Yang-Mills equations was one by mathematicians! The Atiyah-Singer Index Theorem said something about the dimension of the space of solutions to the Yang-Mills equations. The mathematicians had something to say to the physicists, and it is right at this point that the relation between the mathematics and physics community began to change. It is here that the cooperation and cross-fertilization really began. I don’t know how to describe the tremendous difference this made. The entire position that mathematics held within the scientific community changed. The mathematicians actually had something to say!

http://www.ma.utexas.edu/users/lfredrickson/Attachments/TempleLectures.pdf

• See also: http://physics.stackexchange.com/a/1865/37286