Arnold: Swimming Against the Tide by Boris A. Khesin, Serge L. Tabachnikov

By Boris A. Khesin, Serge L. Tabachnikov

Vladimir Arnold, an eminent mathematician of our time, is understood either for his mathematical effects, that are many and widespread, and for his robust reviews, usually expressed in an uncompromising and galvanizing demeanour. His dictum that "Mathematics is part of physics the place experiments are affordable" is celebrated. This booklet comprises components: chosen articles via and an interview with Vladimir Arnold, and a set of articles approximately him written through his buddies, colleagues, and scholars. The booklet is generously illustrated via a wide choice of pictures, a few by no means ahead of released. The publication provides many a side of this remarkable mathematician and guy, from his mathematical discoveries to his daredevil outdoors adventures.

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His innovative method would not have been necessary, he may not have invented it and we may ask whether we would still think that the invariant tori can be constructed only in the real analytic category? Or would somebody else have hit upon Nash’s technique? RECOLLECTIONS 33 My point is that the advances in mathematics require the interaction of many mathematicans, from differerent schools and various countries, as well tools from different fields. The last point is amply clear from Arnold’s lecture.

One might hope — had I conjectured — to find somewhere, at least numerically, an attractor of the Navier–Stokes equation on which the geodesic flow of a negatively curved surface is realized. It was of course very na¨ıve, but I have tried, and in 1964 I made some numerical experiments (with the help of N. Vvedenskaya) on a model with 6 Fourier modes. Unfortunately I was unable to find the positive Lyapunov exponent numerically. At that time, computers produced very-very long tapes with numbers, kilometers of numbers.

Hilbert asked whether by combinations of continuous functions you can get any continuous function in two variables. It is strange, by the way, that Hilbert formulated this problem of algebraic geometry in terms of functions of real variables — but he has done it. In 1956 I was an undergraduate student and Kolmogorov, my supervisor, was working on this problem. He proved that “functions in 4 variables do not exist”: any continuous function in 4 variables or more can be reduced to continuous functions in 3 variables.

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