Serre's thesis and its aftermath rolled in a golden age of algebraic topology which led to the impression that we can really understand (necessarily highly nonlinear) maps from one space to another. With the work of Thom on cobordism and Smale on immersions and the Poincare conjecture, a paradigm developed where geometric problems would be solved by reduction to algebraic topological ones.

The Globally Valued Fields (GVF) project is a joint effort with E. Hrushovski to understand (standard and) non-standard global fields - namely fields in which a certain abstraction of the product formula holds. One possible motivation is to give a model-theoretic framework for various asymptotic distribution results in global fields. Formally, a GVF is a field together with a "valuation" in the additive group of an L^1 space, such that the integral of v(a) vanishes for every non-zero a .

Abstract: We will discuss the geometry of CAT(0) (non-positively curved) cube complexes and groups which act on them. In particular, we will focus on "hyperbolic-like" behavior in such spaces.

I will describe some recent advances in the study of infinite and finite groups, related to probability, growth and complexity. I will start with the celebrated Tits alternative for linear groups, and present extensions and variations, including a joint work with Larsen on a probabilistic Tits alternative. This is related to the notion of probabilistic identities, and related results and open problems will be mentioned. I will then discuss approximate subgroups, an important result by Breuillard-Green-Tao and Pyber-Szabo, and some new growth results with Liebeck and Schul.

Title: Riesz bases of exponentials. Abstract:

The subject of phase transitions and critical phenomena in statistical mechanics is a rich source of interesting and difficult mathematical problems. There has been considerable success in solving such problems for systems in spatial dimension 2, or in high dimensions, but not in dimension 3. This lecture is intended to provide an introduction to recent work that employs a renormalisation group method to study spin systems and self-avoiding walk in dimension 4 (joint with Bauerschmidt and Brydges), as well as long-range versions of these models in dimensions 1,2,3 via an "epsilon expansion."

Abstract: A classical problem in the theory of automorphic forms is to count the number of eigenvalues of the Laplacian acting on cuspidal functions on the quotient of the upper half plane by a lattice R. For R=SL(2,Z) (or a congruence subgroup thereof) the answer is given by Selberg's Weyl law while for the higher rank situation it was established by Mueller and Lindenstrauss-Venkatesh. Additional to the Laplacian there is another large family of operators, namely the Hecke operators attached to SL(2,Z). Sarnak proved that the

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.

Arithmetic Kleinian groups generated by elements of finite order Abstract: We show that up to commensurability there are only finitely many cocompact arithmetic Kleinian groups generated by rotations. The proof is based on a generalised Gromov-Guth inequality and bounds for the hyperbolic and tube volumes of the quotient orbifolds. To estimate the hyperbolic volume we take advantage of known results towards Lehmer's problem. The tube volume estimate requires study of triangulations of lens spaces which may be of independent interest.

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.