Fundamental groups of 3-manifolds are known to satisfy strong
properties, and in recent years there have been several advances in their
study. In this talk I will discuss how some of these properties can be
exploited to give us insight (and results) in the study of 4-manifolds.

We investigate the transversality of holomorphic mappings between CR submanifolds of complex spaces. In equidimension case, we show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target submanifold, provided that the source manifold is of finite type and the map is of generic full rank. In different dimensions, we will show that under certain restrictions on the dimensions and the rank of Levi forms, the mappings whose set of degenerate rank is of codimension at least 2 is transversal to the target. This is a joint work with P. Ebenfelt.

Over the past almost three decades dynamical systems have played a central role in spectral analysis of quasiperiodic Hamiltonians as well as certain quasiperiodic models in statistical mechanics (most notably: the Ising model, both quantum and classical). There are many ways of introducing quasiperiodicity into a model. We shall concentrate on the widely studied Fibonacci case (which is a prototypical example of so-called substitution systems on two letters with certain desirable properties). In this case a particular geometric scheme, arising from a certain smooth three-dimensional dynamical system associated to the quasiperiodic model in question (the so-called Fibonacci trace map) has been established. Our aim is to present a general dynamical/geometric framework and to demonstrate how information about the model in question (spectral properties for Hamiltonians, and Lee-Yang zeros distribution for classical Ising models) can be obtained from the aforementioned dynamical system and the geometry of certain dynamically invariant sets. In this first in a series of two (or three) talks, we'll briefly recall how dynamical systems are associated to Schroedinger and Jacobi operators, as well as classical Ising models. We'll establish notation, ask main questions and in general prepare the ground for a somewhat more general (in terms of geometry and dynamical systems) discussion for next time.

We complete the construction of the Martin-Solovay tree