Abstract: There are by now several celebrated measure classification results to the effect that a measure is uniform provided it possesses sufficient "invariance" as quantified by stabilizer, entropy, or recurrence. In some applications, part of the challenge is to identify or construct measures to which these hypotheses apply.

In the paper "Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math 89 (1999) 227-262", Lindenstrauss showed that minimal systems of mean dimension less than $cN$ for $c=1/36$ embed equivariantly into the Hilbert cubical shift $([0,1]^N)^{\mathbb{Z}}$, and asked what is the optimal value for $c$. We solve this problem by proving that $c=1/2$. The method of proof is surprising and uses signal analysis sampling theory. Joint work with Masaki Tsukamoto.

Let $A$, $B$ be two rational functions of degree at least two on the Riemann sphere. The function $B$ is said to be semiconjugate to the function $A$ if there exists a non-constant rational function $X$ such that the equality (*) A\circ X=X\circ B holds. The semiconjugacy relation plays an important role in the classical theory of complex dynamical systems as well as in the new emerging field of arithmetic dynamics. In the talk we present a description of solutions of (*) in terms of two-dimensional orbifolds of non-negative Euler characteristic on the Riemann sphere.

20 years ago Benjy Weiss constructed a collection of non-trivial translation invariant probability measures on the space of entire functions. In this talk we will present a construction of such a measure, and give upper and lower bounds for the possible growth of entire functions in the support of such a measure. We will also discuss "uniformly recurrent" entire functions, their connection to such constructions, and their possible growth. The talk is based on a joint work with Lev Buhovski, Alexander Loganov, and Mikhail Sodin.

We give a brief overview on applications of the Poincare's equation to the study of random walk on the the Sierpi ́nski gasket. In particular, we discuss such questions as anomalous diffusion, relation to branching processes and decimation invariance. Metods of the complex analysis and the iteration theory are used to deal with the aforemen-tioned problems.

Abstract: We will present an elementary problem and a conjecture regarding percolation on planar graphs suggested by assuming quasi invariance of percolation crossing probabilities under coarse conformal uniformization.

Let S be a finite set (the sample space), and  f_i: S -> R functions, for 1 ≤ i ≤ k. Given a k-tuple (v_1,...,v_k) in R^k it is natural to ask:  What is the distribution P on S that maximizes the entropy       -Σ P(x) log(P(x)) subject to the constraint that the expectation of f_i be v_i? In this talk I'll discuss a closed formula for the solution P in terms of a sum over cumulant trees. This is based on a general calculus for solving perturbative optimization problems due to Feynman, which may be of interest in its own right. 

Abstract: This talk will introduce the notion of Gaussian and almost Gaussian log-correlated fields. These are a class of random (or almost random) functions many of whose statistics are predicted to coincide in a large system-size limit. Examples of these objects include: (1) the logarithm of the Riemann zeta function on the critical line (conjecturally) (2) the log-characteristic polynomial of Haar distributed unitary random matrices (and others), (3) the deviations of Birkhoff sums of substitution dynamical systems (conjecturally)

Title: Local limit theorem for certain ballistic random walks in random environments Abstract: We study the model of random walks in random environments in dimension four and higher under Sznitman's ballisticity condition (T'). We prove a version of a local Central Limit Theorem for the model and also the existence of an equivalent measure which is invariant with respect to the point of view of the particle. This is a joint work with Noam Berger and Moran Cohen.

Title: Self-affine measures with equal Hausdorff and Lyapunov dimensions Abstract: Let μ be the stationary measure on ℝd which corresponds to a self-affine iterated function system Φ and a probability vector p. Denote by A⊂Gl(d,ℝ) the linear parts of Φ. Assuming the members of A contract by more than 12, it follows from a result by Jordan, Pollicott and Simon, that if the translations of Φ are drawn according to the Lebesgue measure, then dimHμ=min{D,d} almost surely. Here D is the Lyapunov dimension, which is an explicit constant defined in terms of A and p.