In the first part of the talk I will introduce the Becker-Doring
nucleation equation and describe its singular perturbation solution under
time-dependent conditions of a nucleation pulse. In the second part, I
will discuss a supersaturated lattice gas on a square lattice, where
steady-state and time-dependent nucleation can be described from first
principles. Comparison confirms qualitative (not quantitative) validity of
the Becker-Doring model at not too small temperatures T , but also reveals
its limitations due to neglect of "magic numbers", which become prominent
as T -> 0 .

We show that given $\omega$ many supercompact cardinals, there is a
generic extension in which there are no Aronszajn trees at
$\aleph_{\omega+1}$. This is an improvement of the large cardinal
assumptions. The previous hypothesis was a huge cardinal and $\omega$ many
supercompact cardinals above it, in Magidor-Shelah.

How to extract trend from highly nonlinear and nonstationary data is an important problem that has many practical applications ranging from bio-medical signal analysis to econometrics, finance, and geophysical fluid dynamics. We review some exisiting methodologies in defining trend and instantaneous frequency in data analysis. Many of these methods use pre-determined basis and is not completely adaptive. They tend to introduce artificial harmonics in the decomposion of the data. Various attempts to preserve the temportal locality property of the data introduce problems of their own. Here we discuss how adaptive data analysis can be formulated as a nonlinear optimization problem in which we look for a sparse representation of data in some unknown basis which is derived from the physical data. We will show that this formulation has some beautiful mathematical structure and can be considered as a nonlinear version of compressed sensing.

We propose efficient Eulerian methods for approximating the
finite-time Lyapunov exponent (FTLE). The
idea is to compute the related flow map using the level set method and
the Liouville equation. There are
several advantages of the proposed approach. Unlike the usual
Lagrangian-type computations, the resulting
method requires the velocity field defined only at discrete locations.
No interpolation of the velocity field
is needed. Also, the method automatically stops a particle trajectory
in the case when the ray hits the
boundary of the computational domain. The computational complexity of
the algorithm is O(1/x^(d+1))
with d the dimension of the physical space. Since there are the same
number of mesh points in the x-t space,
the computational complexity of the proposed Eulerian approach is
optimal in the sense that each grid
point is visited for only O(1) time. We also extend the algorithm to
compute the FTLE on a co-dimension
one manifold. The resulting algorithm does not require computation on
any local coordinate system and is
simple to implement even for an evolving manifold.