According to DiPerna-Lions theory, velocity fields with weak
derivatives in L^ p   spaces possess weakly regular flows. When a velocity
field is perturbed by a white noise, the corresponding (stochastic) flow
is far more regular in spatial variables; a diffusion with a drift in a
suitable L p   space possesses weak derivatives with exponential bounds.
As an application we show that a Hamiltonian system that is perturbed by a
white noise produces a symplectic flow for a Hamiltonian function that is
merely in W^{ 1,p}  for p  strictly larger than dimension. I also discuss
the potential application of such regularity bounds to study solutions of
Navier-Stokes equation with the aid of Constantin-Iyer's circulation
formula.

We present  conditional existence results for  the Landau equation with
Coulomb potential. Despite lack of a comparison principle for the equation, the
proof of existence relies on barrier arguments and parabolic regularity theory. The
Landau equation arises in kinetic theory of plasma physics. It was derived by Landau
and serves as a formal approximation to the Boltzmann equation when grazing
collisions are predominant. 
We also present long-time existence results for the isotropic version of the Landau
equation with Coulomb potential.

Homogeneity of closed sets was introduced by Carleson in a 1983 paper which solved the Corona problem on a general class of domains in the complex plane. Recent results of several authors have shed light on the importance of homogeneity from the point of view of inverse spectral theory. I will present some recent work which constructs several large classes of limit periodic operators whose spectra are Carleson-homogeneous Cantor sets.

I will discuss some analogies between the second main

theorem in Nevanlinna's theory and results in holomorphic dynamics.

The two main examples will be equidistribution results for endomorphisms of $\P^k$ and

equidistribution results for singular foliations by Riemann-surfaces in $\P^2$.

This is joint work with T. C. Dinh.

Abstract: We consider the nonlinear instability of a steady state
$v_*$ of the Euler equation in an $n$-dim fixed smooth bounded domain. When
considered in $H^s$, $s>1$, at the linear level, the stretching of the
steady fluid trajectories induces unstable essential spectrum which
corresponds to linear instability at small spatial scales and the
corresponding growth rate depends on the choice of the space $H^s$.
More physically interesting linear instability relies on the unstable
eigenvalues which correspond to large spatial scales. In the case when
the linearized Euler equation at $v_*$ has an exponential dichotomy of
center-stable and unstable (from eigenvalues) directions, most of the
previous results obtaining the expected nonlinear instability in $L^2$
(the energy space, large spatial scale) were based on the vorticity
formulation and therefore only work in 2-dim. In this talk, we prove,
in any dimensions, the existence of the unique local unstable manifold
of $v_*$, under certain conditions, and thus its nonlinear
instability. Our approach is based on the observation that the Euler
equation on a fixed domain is an ODE on an infinite dimensional
manifold of volume preserving maps in function spaces. This is a
joint work with Zhiwu Lin.