Abstract: Renormalization provides a powerful tool to approach universality and
rigidity phenomena in dynamical systems. In this talk, I will discuss
recent results on renormalization and rigidity theory of circle
diffeomorphisms (maps) with a break (a single point where the derivative
has a jump discontinuity) and their relation with generalized interval
exchange transformations introduced by Marmi, Moussa and Yoccoz. In a
joint work with K.Khanin, we proved that renormalizations of any two
sufficiently smooth circle maps with a break, with the same irrational
rotation number and the same size of the break, approach each other
exponentially fast. For almost all (but not all) irrational rotation
numbers, this statement implies rigidity of these maps: any two
sufficiently smooth such maps, with the same irrational rotation number
(in a set of full Lebesgue measure) and the same size of the break, are
$C^1$-smoothly conjugate to each other. These results can be viewed as
an extension of Herman's theory on the linearization of circle
diffeomorphisms.
 

 The Ginzburg - Landau equations play a fundamental role in various areas of physics, from  superconductivity to elementary particles. They present the natural and simplest extension of the Laplace equation to line bundles. Their non-abelian generalizations - Yang-Mills-Higgs and Seiberg-Witten equations have applications in geometry and topology. 

      Of a special interest are the least energy (per unit volume) solutions of the Ginzburg - Landau equations. These turned out to have a beautiful structure of (magnetic) vortex lattices discovered by A.A. Abrikosov. (Their discovery was recognized a Nobel prize. Finite energy excitations are magnetic vortices, called Nielsen-Olesen or Nambu strings, in the particle physics.)

      I will review recent results about the vortex lattice solutions and their relation to the energy minimizing solutions on Riemann surfaces and, if time permits, to the microscopic (BCS) theory.

Toeplitz sequences are constructed from periodic sequences with undetermined positions by successively filling these positions with the letters of other periodic sequences. In this talk we will consider  the class of so called simple Toeplitz sequences. We will describe combinatorial properties, such as the word complexity, of the subshifts that are associated with them. The relation between combinatorial properties of the coding sequences and the Boshernitzan condition will be also discussed.

The Favard length of a set E has a probabilistic interpretation: up to a constant factor, it is the probability that the Buffon's needle, a long line segment dropped at random, hits E. In this talk, we study the Favard length of some random Cantor sets of dimension 1. Replace the unit disc by 4 disjoint sub-discs of radius 1/4 inside. By repeating this operation in a self-similar manner and adding a random rotation in each step, we can generate a random Cantor set D. Let D_n be the n-th generation in the construction, which is comparable to the 4^{-n}-neighborhood of D. We are interested in the decay rate of the Favard length of these sets D_n as n tends to infinity, which is the likelihood (up to a constant) that the Buffon's needle will fall into the 4^{-n}-neighborhood of D. It is well known that the lower bound for such 1-dimensional set is constant multiple of 1/n. We show that the upper bound of the Favard length of D_n is also constant multiple of 1/n in the average sense.