What is the KAM project?

KAM is an acronym for Kolmogorov-Arnold-Moser. KAM theory gets its origins back to the classical work on celestial mechanics and perturbation theory. In the Keplerian motion, one neglects the influence of other planets. Say one may think that the moon is rotating around the Earth but this motion does not take into account the influence of the sun on it. This is fine for small time but not for large scale of times. The interaction can also be used to detect a planet: the anomaly in the motion of Uranus lead to the discovery by Galle and Le Verrier of Neptune.

Celestial mechanics became a classical subject of mathematics and gave eventually birth to diferential calculus and later to quantum mechanics and perturbation theory. In the 50's the subject had been abandoned, physics was more concerned with quantum field theories than old celestial mechanics. Nevertheless Kolmogorov discovered the existence of stable quasi-periodic motions and it was confirmed by Arnold that these exists in the planar three body problem.

Kolmogorov basic idea is rather than solving differential equations search manifolds which are invariant, the KAM tori. Seventy years later, the subject develop and became fascinating. But... as it grew, it became more and more technical and hermetic. Only the very happy few, who can master pages of estimates are admitted within the small circle of experts, even among mathematicians working in dynamical systems.

One can hardly imagine an algebraic geometer not knowing the Riemann Roch theorem in its simplest form or the Hodge decomposition theorem. It is customary to meet specialist in dynamical system having only superficial knowledge in KAM theory.

Now imagine for a moment that algebraic geometry develpped without commutative algebra, without sheaf theory and without cohomology then you will get a picture of the situation we face in KAM theory. The KAM project aims at establishing foundations for KAM theory. Of course, these might seem ridiculous to many experts who do not need any theory but we think it is not.

While KAM theory is sometimes understood in a narrow sense as the study of invariant tori, we share a different point of view: we consider KAM theory as a part of category theory, namely the theory of Banach valued functors defined on a small category. This branch of functional analysis we propose to call functorial analysis.