The index theorem, at the top

American mathematician Isadore Singer died on February 11, at the age of 96. With his collaborator Michael Atiyah, who died in 2019, he had proved the index theorem, famous among mathematicians, which earned them the Abel Prize in 2004. The exceptional importance of this theorem is attested by the fact that it establishes an unsuspected link between two parts of mathematics that were previously distant, analysis and topology, but also by its consequences in theoretical physics. We often think, wrongly, that the role of the mathematician consists in solving equations. In fact, there are all kinds of equations. Many of those encountered in physics involve unknowns that are functions rather than numbers. These are called differential equations and their study is part of “mathematical analysis”. It is rare that we know how to solve this type of equation, but the index theorem allows us to count the number of their solutions, which is often sufficient for applications. Atiyah and Singer associate to the equation an object called a “fibered”, the study of which is part of topology, and on which one can directly read the number of solutions. A bridge is thus established between analysis and topology.
The theorem was proved in 1963 but Atiyah and Singer did not publish a proof until 1968. In fact, they waited until they had three different proofs, a bit like a summit that one reaches by several ways, each one bringing a new perspective. All this did not suddenly appear in their minds. For more than twenty years, they developed their ideas based on many previous theorems that did not seem to be related. The most important advances in mathematics are often syntheses: heterogeneous results suddenly appear as mere special cases of a much more powerful theory.

The external and the internal

A few years later, the link with physics became clear. The “gauge theory” of physicists was very close to the “fibers” of mathematicians. The index theorem became a crucial tool in quantum physics. It can be seen as an example of the “unreasonable efficiency of mathematics in the natural sciences”, to use a famous expression of the physicist Eugene Wigner.

The links between physics and mathematics are as old as science, and opinions differ. The mathematician Vladimir Arnold asserted that mathematics is only a chapter of physics. Others insist instead on the importance of mathematics as an abstract and autonomous discipline. The view of Atiyah and Singer is intermediate. According to them, almost all mathematics was born from external reality, for example what concerns numbers, but then it turned to internal questions, such as the theory of prime numbers. Other parts of mathematics, on the other hand, are closer to the external world and physics plays a crucial motivating role. The strength of mathematics lies in these two complementary components: external and internal. In 1900, David Hilbert stated that “a mathematical theory can only be considered complete if it is so clear that you can explain it to the first person you meet in the street”. Alas, we will have to wait a little longer before we can clearly explain the index theorem to the readers of Le Monde!