We consider the global embedding problem for compact, three dimensional
CR manifolds. Sufficient conditions for embeddability are obtained from assumptions on the CR Yamabe
invariant and the non-negativity of a certain conformally invariant fourth order operator called the CR Paneitz
operator. The conditions are shown to be necessary for small deformations of the standard CR structure on the three sphere.

Last time we saw how dynamical systems are associated to certain quasiperiodic models in physics. We also saw the need for a general investigation of dynamics of trace maps and the geometry of some dynamically invariant sets, motivating this week's discussion. We'll investigate in greater generality dynamics of the Fibonacci trace map, geometry of so-called stable manifolds, and we'll see how this information can be used to get detailed topological, measure-theoretic and fractal-dimensional description of spectra of quasiperiodic (Fibonacci) Schroedinger and Jacobi Hamiltonians, as well as the distribution of Lee-Yang zeros for the classical Ising model. Time permitting, we'll also mention recent applications in the theory of orthogonal polynomials.

In this series of two talks I will give an introduction to some of my recent research on the ineffable tree property. The ineffable tree property is a two cardinal combinatorial principle which can consistently hold at small cardinals. My recent work has been on generalizing results about the classical tree property to the setting of the ineffable tree property. The main theorem that I will work towards in these talks generalizes a theorem of Cummings and Foreman. From omega supercompact cardinals, Cummings and Foreman constructed a model where the tree property holds at all of the $\aleph_n$ with $1 < n < \omega$. I recently proved that in their model the $(\aleph_n,\lambda)$ ineffable tree property holds for all $n$ with $1 < n < \omega$ and $\lambda \geq \aleph_n$.