We consider random products of SL(2, R) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense $G_\delta$ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrodinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice. This is a joint project with A.Gorodetski.

 Three different math stories in one lecture. Only definitions, motivations, results, some ideas behind proofs, open questions. 

1. One of the greatest achievements in Dynamics in the XX century is the KAM Theory. It says that a small perturbation of a non-degenerate completely integrable system still has an overwhelming measure of invariant tori with quasi-periodic dynamics. What happens outside KAM tori remains a great mystery. The main quantitate invariants so far are entropies.  It is easy, by modern standards, to show that topological entropy can be positive. It lives, however, on a zero measure set. We are now able to show that metric entropy can become positive too, under arbitrarily small C^{infty} perturbations, answering an old-standing problem of Kolmogorov. Furthermore, a slightly modified construction resolves another long–standing problem of the existence of entropy non-expansive systems. In these modified examples  positive metric entropy is generated in arbitrarily small tubular neighborhoods of one trajectory. Joint with S. Ivanov and Dong Chen.

2. A survival guide for a feeble fish and homogenization of the G-Equation. How fish can get from A to B in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water in the ocean? This is related to the G-Equation and has applications to its homogenization. The G-equation is believed to govern many combustion processes, say wood fires or combustion in combustion engines (generally, in pre-mixed media with “turbulence".  Based on a joint work with S. Ivanov and A. Novikov.

3. Just 20 years ago the topic of my talk at the ICM was a solution of a problem which goes back to Boltzmann and has been formulated mathematically by Ya. Sinai. The conjecture of Boltzmann-Sinai states that the number of collisions in a system of $n$ identical balls colliding elastically in empty space is uniformly bounded for all initial positions and velocities of the balls. The answer is affirmative and the proven upper bound is exponential in $n$. The question is how many collisions can actually occur. On the line, one sees that  there can be $n(n-1)/2$ collisions, and this is the maximum. Since the line embeds in any Euclidean space, the same example works in all dimensions. The only non-trivial (and counter-intuitive) example I am aware of is an observation by Thurston and Sandri who gave an example of 4 collisions between 3 balls in $R^2$. Recently, Sergei Ivanov and me proved that there are examples with exponentially many collisions between  $n$ identical balls in $R^3$, even though the exponents in the lower and upper bounds do not perfectly match. Many open questions left.

The study of the equation on the 2-torus given by  
x’= sin x + a + b sin t
has been motivated by its relation to the Josephson junction in physics, as well as by purely mathematical reasons. For any values of the parameters a and b, one can consider the time-2\pi (period) map from the x-circle to itself, and study its properties, in particular, its rotation number.

Study of the Arnold tongues corresponding to this family, reveals a miracle: sometimes, their left and right boundaries intersect at a hourglass-type so-called adjacency point. Moreover, the a-coordinates of all these points turn out to be integers. My talk will be devoted to the geometry behind all of this, summarizing the works of many authors: Ilyashenko, Guckenheimer, Buchstaber, Karpov, Tertychnyi, Glutsyuk, Klimenko, Schurov, Filimonov, Romaskevich, Ryzhov, and myself.
 

We will discuss several open problems in dynamical systems (related to random dynamical systems with parameters, sums of Cantor sets, etc.) that can potentially turn into a research project suitable for graduate students.