We discuss the continuous time percolation model in an ergodically defined
environment. Under minimal assumptions on the ergodic system, we show the
existence of sets of sampling functions with percolation or extinction
showing that the latter are dense open. We also discuss the related
spatially inhomogeneous continuous time random cluster model and the
topological properties of sets of sampling functions corresponding to
percolation and decay.

Magnetic resonance images typically contain signals from multiple chemical species such as water and fat. The diagnostic information in the image can be improved by separating the components of the signal coming from individual chemical species. The model that describes the signal generation includes non-linear parameters which arise from imperfections in the magnetic field and signal decay. The Cramer-Rao Bound is the minimum variance of an unbiased estimator of a parameter. In this work, we use the Cramer-Rao Bound to optimize the data acquisition for the non-linear inverse problem of estimating the magnetic field inhomogeneities and signal decay.

The tree property at \kappa^+ states that every tree with height \kappa^+ and levels of size at most \kappa has an unbounded branch. There is a tension between the tree property and the Singular Cardinal Hypothesis (SCH). Woodin asked if the failure of SCH at a singular cardinal \kappa implies the tree property at \kappa^+. Recently Neeman answered this question in the negative. Here we show that this result can be obtained at small cardinals. In particular we will show that given \omega many supercompact cardinals there is a generic extension in which the tree property holds at \aleph_{\omega^2+1} and SCH fails at \aleph_{\omega^2}.

One of the most difficult problems in mathematics and physics is to
find an accurate, practical description of turbulent flows. Turbulence
is ubiquitous in nature, occurring in very diverse physical settings,
such as aerodynamics, geophysics, weather and climate modeling, ocean
and atmospheric flows, star formation, blood flow in the heart, and
many others. This problem is not only untenable by current
mathematical tools, but direct numerical simulation of detailed
turbulent flows has proven to be computationally prohibitive, even
using the most powerful state-of-the-art computers. A major piece of
the puzzle of understanding these phenomena is widely believed to lie
in a system of nonlinear PDEs known as the Navier-Stokes equations,
which are the subject of one of the seven $1,000,000 Clay Millennium
Prize problems. I will discuss give an introduction to the
Navier-Stokes equations and discuss their relationship to turbulence
and the Millennium problem.