I will discuss the role that independence relations play in modern model theory, discussing the classes of stable, simple, and rosy theories along the way. I will then discuss why the Urysohn space is not
stable or simple, but is rosy. Part of the talk reflects joint work with Clifton Ealy.

Plane waves for two phase flow in a porous medium are modeled by the one-dimensional Buckley-
Leverett equation, a scalar conservation law. In the first part of the talk, we study traveling wave solutions of the equation modfied by the Gray-Hassanizadeh model for rate-dependent capillary pressure. The modfication adds a BBM-type dispersion to the classic equation, giving rise to under-compressive waves. In the second part of the talk, we analyze stability of sharp planar interfaces (corresponding to Lax shocks) to two-dimensional perturbations, which involves a system of partial differential equations. The Safman-Taylor analysis predicts instability of planar fronts, but their
calculation lacks the dependence on saturations in the Buckley-Leverett equation. Interestingly, the dispersion relation we derive leads to the conclusion that some interfaces are long-wave stable and some are not. Numerical simulations of the full nonlinear system of equations, including dissipation and dispersion, verify the stability predictions at the hyperbolic level. This is joint work with Kim Spayd and Zhengzheng Hu.

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.