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 trendy, and for his robust critiques, frequently expressed in an uncompromising and inspiring demeanour. His dictum that "Mathematics is part of physics the place experiments are reasonable" is widely known. This publication contains elements: chosen articles by way of and an interview with Vladimir Arnold, and a suite of articles approximately him written by way of his acquaintances, colleagues, and scholars. The e-book is generously illustrated via a wide choice of images, a few by no means earlier than released. The ebook provides many a side of this remarkable mathematician and guy, from his mathematical discoveries to his daredevil outside adventures.
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Extra resources for Arnold: Swimming Against the Tide
We thus get a dynamical system on this curve. We have two involutions of the curve: the vertical involution A and the horizontal involution B and hence we have a mapping T = AB, a diﬀeomorphism of the circle preserving the orientation. After some experiments I was able to ﬁnd that for these diﬀeomorphisms the rotation number exists, there might be resonances, periodic orbits and so on. Then I found that Poincar´e had already studied diﬀeomorphisms of the circle onto itself preserving the orientation and created a theory for this.
The discriminant hypersurface is the set of all points where the function z is not nice, in particular, not smooth. In the 60’s, around 1967, I started to think about how to use the topology of this object to deduce from it an obstacle to the representation of algebraic functions in terms of algebraic functions of a smaller number of variables. I thought that the topology of our algebraic function for higher n is complicated and, if there were an expression in terms of functions of fewer variables, then it should be simpler.
The proof was based on a technology which he called the theory of multidimensional variations and which is in fact a version of integral geometry of the Chern classes describing the integrals over cycles in Grassmann varieties. His technology was based on some evaluations of topological complexity in real algebraic geometry. This is also one of the main problems in mathematics. In the simplest case, for the curves, you have a polynomial equation in 2 variables, say of degree n, and you want to know the topology of the variety deﬁned by this equation (in higher dimensions, by a system of such equations in the aﬃne or projective space).