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. 

A (countable discrete) group $\Gamma$ acting on a compact space $X$ is said to act \emph{amenably} if there is a continuous net $(\mu_n^x)$ of probability measures indexed by the points of $X$ that are almost invariant under the action of $\Gamma$. For example, $\Gamma$ is amenable if and only if it acts amenably on a one-point space. The protoypical example of a boundary amenable non-amenable group is a non-abelian free group. More generally, if acts properly, isometrically, and transitively on a tree, then $\Gamma$ is boundary amenable. In this talk, I will present a construction of the Stone-Cech compactification of a locally compact space using C*-algebra ultrapowers that allows one to give a slick proof of the aforementioned result. This construction is motivated by the open question as to whether or not Thompson’s group is boundary amenable and I will also discuss the optimistic thought that this construction could be used to settle that problem.

In the classical Polya’s urn process, there are balls of different colors in the urn, and one step of the process consists of taking out a random ball and it putting back together with one more ball of the same color. ("Ask a friend whether he’s using is A or B, and buy the same".)

It also can be modified by saying that the reinforcement probability is proportional not to the number of balls of a given color, but to its power $\alpha$, and the asymptotic behaviour for $\alpha=1$, $\alpha>1$ and for $\alpha<1$ are quite different.

The talk will be devoted to the model of interacting urns : at each moment, there is a competition for reinforcement between randomly chosen subset of colours; for a real-life analogue, one can consider companies competing on different markets (one company produces toys and computers, another sells computers and cars, etc.).

We will describe possible types of the limit behaviour of such model for different values of $\alpha$; it turns out that what happens for $\alpha>>1$ is quite different from $\alpha=1$, and both are quite interesting (this is a joint work with Christian Hirsch and Mark Holmes).

We discuss an inverse theorem on the structure of pairs of discrete probability measures which has small amount of growth under convolution, and apply this result to self-similar sets with overlaps to show that if the dimension is less than the generic bound, then there are superexponentially close cylinders at all small enough scales. The results are by M.Hochman.