Latest of fathers of KAM theorem, Vladimir Arnold, passed away yesterday. He was 72. He has been one of the most influential mathematical-physicists of XX century. I studied analytical mechanics on his book and I have learned a lot about differential equations from other fine texts of him. It is with great sadness for me to apprehend this piece of news that is going around all the World, giving the exact stature of this man. The reason for my deep sadness relies on the fact that I built on his very strong shoulders, being him a giant (see here). KAM theorem has been a key result in dynamical systems opening the way toward a deep understanding of chaos in conservative systems and extending beyond to more general classes of Hamiltonian systems. Personally, I lived the struggling to understand plasma heating under the spell of this theorem but applications range everywhere in all fields of physics. I think that the best way to remember a scientist is through his works and I think that a simple post is not enough to embrace all the contributions due to Arnold. He is also remembered for being a pioneer in catastrophe theory and solved XIII Hilbert problem. He has been awarded a Fields medal but Soviet government impeded him to accept the prize. He will live forever in the work of people of our community and a lot more will be the fruits coming from his ideas.

## Arnold passed away

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At least according to Wikipedia, which may or may not be correct, or may depend on the editor ;), the

XIII-th Hilbert problem is still unresolved.

“A variant of this problem, searching for a solution within the universe of continuous functions, was solved by Andrei Kolmogorov and Vladimir Arnold. It is not difficult to show that the problem has a positive solution within the space of single-valued analytic functions (Raudenbush). Some authors argue that Hilbert intended for a solution within the space of (multi-valued) algebraic functions, thus continuing his own work on algebraic functions and being a question about a possible extension of the Galois theory (see, for example, Abhyankar,[14] Vitushkin,[15] Chebotarev [16] and others). It appears from one of Hilbert’s papers [17] that this was his original intention for the problem.

The language of Hilbert there is “…Existenz von algebraischen Funktionen…”, i.e., “…existence of algebraic functions…”. As such, the problem is still unresolved.”

(http://en.wikipedia.org/wiki/Hilbert's_problems)

But in the section “Summary” I find

“Of the cleanly-formulated Hilbert problems, problems 3, 7, 10, 11, 13, 14, 17, 19, 20, and 21 have a resolution that is accepted by consensus. On the other hand, problems 1, 2, 5, 9, 15, 18+, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether it resolves the problem.”

I think some Wikipedia editor should properly fix this. Arnold is claimed to have solved this problem.

Marco