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.