We present several geometric interpretations of a certain family of
solutions of an "integrable" nonlinear pde. This sheds light on a diverse
range of topics, from the classical Painleve equations to the
quantum cohomology of Fano manifolds.

This talk will focus on orthogonal polynomials whose corresponding measure of orthogonality is not supported on the real line or unit circle. In this setting, the orthonormal polynomials do not satisfy a three-term recurrence relation. However, many theorems from the classical settings of the real line and unit circle can be reformulated to apply to this more general situation. The first part of this talk will present some history and motivation for studying these polynomials and we will conclude by presenting some new results.

In the previous two talks we established a dictionary between some properties of quasiperiodic (particularly Fibonacci) models and some geometric constructions arising as dynamical invariants for the Fibonacci trace map. In this talk we shall apply our findings to a specific model: the classical 1D Ising model with quasiperiodic magnetic field and quasiperiodic nearest neighbor interaction. In particular, we'll prove absence of phase transitions of any order and we'll investigate the structure of Lee-Yang zeroes in the thermodynamic limit (these are zeroes of the partition function as a function of the complexified magnetic field---while in finite volume the partition function is a polynomial whose zeroes fall on the unit circle, a challenge is to determine whether in infinite volume (thermodynamic limit) these zeroes accumulate on any set on the unit circle, and if so, to determine the structure of this set). The purpose of this work is to serve as rigorous justification to previously observed phenomena (mostly through numerical and some soft analysis). Should we have time, we'll also very briefly mention applications of the aforementioned dictionary to quasiperiodic Jacobi matrices/CMV matrices.

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.