The Herman conjecture is probably a cornerstone in KAM theory. In his ICM lecture of 1998, Michel Herman formulated several problems and conjecture among which the invariant tori conjecture. He did it for discrete dynamics but there is also a conjecture for continuous dynamics and we consider the later one for simplicity. The weird thing is that the conjecture is in some sense the simplest version possible of the KAM theorem: it states that, under arithmetic conditions, near an equilibrium which is stable at the linear level, Hamiltonian motion has a positive measure set of invariant tori carrying quasi-periodic motions. A non specialist will hardly notice the difference with the KAM theorem.
The question one might ask is: "how are we able to prove the KAM theorem and not be able to solve the Herman conjecture?"
In our point view, this was due to our absence of a deep understanding of the standard KAM theorem. In 2012, we proposed a proof in three steps:
Introduce an appropriate normal form to replace the old Birkhoff normal form.
Construct an abstract KAM theorem to prove convergence towards this normal form over a Cantor set.
Show that the Cantor set has positive measure.
Specialists did not accept he proof, mostly considered as a fake. This lead to reconsider each part of the proof in detail and make it more and more transparent.
The talk given by van Straten at CIRM in 2021 gives an introduction to the Herman conjecture.
The link to our ArXiv paper is:
https://arxiv.org/abs/1909.06053
In 2022, we held a zoom seminar recorded them, explaining the proof. It is sufficient simple that good undergraduates student can understand it, so somehow we accomplished what we wanted get back high mathematics to a very basic level in algebra,functional analysis and geometry.