My talk will be devoted to a joint work with M. Katsnelson, A. Okunev, I. Schurov and D. Zubov.

Graphene is a layer of carbon (forming a hexagonal lattice) of thickness of one or several atoms. One of its remarkable properties is that the behavior of electrons on it is described by the Dirac equation, the same equation that describes the behavior of ultrarelativistic particles. A corollary of this is the Klein tunneling: an electron (or, as it is much more appropriate to say, an wave or quasiparticle) that falls orthogonally on a flat potential barrier on a single-layer graphene, not only has a positive chance of tunneling through it (what is quite natural in quantum mechanics), but passes through it with probability one(!).

Reijnders, Tudorovskiy and Katsnelson, while modeling a transition through an n-p-n junction, have discovered the presence of other, nonzero "magic" angles, under which the falling particle (of given energy) passes through the barrier with probability one.

There are a several interesting problems that arise out of this work. On the one hand, a zero probability of reflection is a codimension two condition (the coefficient before the reflected wave is a complex coefficient that should be equal to zero). Thus, we have a system of two equations on one variable (the incidence angle) that has nonempty set of solutions, what one would not normally expect. And it is interesting to explain their origins.

On the other hand, there is a question that is interesting from the point of view of potential applications: can one invent a potential that "closes well" the transition probabilities (in particular, that has no magic angles)? This question comes from construction of transistors: that is what we should observe for a transistor in the "closed" state.

I will speak about our advances in all these problems. In particular, the tunneling problem on bilayer graphene turns out to be (vaguely) connected to the slow-fast systems on the 2-torus.

As part of a more general conjecture by Katok and Spatzier, it was asked if all smooth Anosov Z^r-actions on tori, nilmanifolds and infranilmanifolds without rank-1 factor actions are, up to smooth conjugacy, actions by automorphisms. In this talk, we will discuss a recent joint work with Federico Rodriguez Hertz that affirmatively answers this question.

A number is called badly approximable if there is a positive constant c such that |x-p/q| > c/q^2 holds for all rationals p/q, so that close approximation by rationals requires relatively large denominators. The set of such numbers is Lebesgue-null but has full Hausdorff dimension. This set can be viewed as the union over c of the set BA(c) of numbers which satisfy the above inequality for the fixed constant c. J. Kurzweil obtained dimension bounds on BA(c), which were later improved by D. Hensley. We will discuss recent work, joint with D. Kleinbock, in which we use homogeneous dynamics to produce dimension bounds for a higher-dimensional analog.

Since the early 1970's, it has been known in both, the mathematical physics and in the physics communities, that propagation of information in quantum spin chains cannot exceed the so-called Lieb-Robinson bound (effectively providing the quantum analog of the light cone from the relativity theory). Typically these bounds depend on the parameters of the model (interaction strength, external field). The recent Hamza-Sims-Stolz result demonstrates exponential localization (a la Anderson localization) of information propagation in most spin chains (in the sense of a given probability distribution with respect to which interaction and external field couplings are drawn). A natural question arises: what can be said about lower bounds on propagation of information in spin crystals (i.e. the case far from the one in which localization is expected), as well as in the intermediate case--the spin quasicrystals. This problem can be reduced to solving a linear ODE given by a Hermitian matrix, the solutions of which live on finite-dimensional complex spheres.

In this talk we shall discuss the history, give a general overview of the field, reduce the problem to an ODE problem as mentioned above, and look at some open problems. We shall also present some numerical computations with animations.