Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing. In this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time. In this lecture I will discuss the discrete data assimilation for the incompressible two-dimensional Navier-Stokes
equations. In both cases we obtain bounds on the time interval between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution.

BALLISTICITY CONDITIONS FOR RANDOM WALK IN
RANDOM ENVIRONMENT
ALEJANDRO F. RAMIREZ
resumen. Consider a Random Walk in a Random Environment (RWRE)
{Xn :
n
≥ 0} on a uniformly elliptic i.i.d. environment in dimensions d ≥ 2. Some
fundamental questions about this model, related to the concept of ballisticity
and which remain unsolved, will be discussed in this talk. The walk is said to
be transient in a direction l
∈ S
d
, if limn
→∞ Xn l = ∞, and ballistic in the
direction l if lim inf n
→∞ Xn l/n > 0. It is conjectured that transience in a
given direction implies ballisticity in the same direction. To tackle this question,
in 2002, Sznitman introduced for each γ
∈ (0, 1) and direction l the ballisticity
condition (Tγ )
|l, and condition (T ′ )|l defined as the fulfillment of (Tγ )|l for each
γ
∈ (0, 1). He proved that (T ′ ) implies ballisticity in the corresponding direction,
and showed that for each γ
∈ (0, 5, 1), (Tγ ) implies (T ′ ). It is believed that for
each γ
∈ (0, 1), (Tγ ) implies (T ′ ). We prove that for γ ∈ (γd , 1), (T )γ is equivalent
to (T ′ ), where for d
≥ 4, γd = 0 while for d = 2, 3 we have γd ∈ (0.366, 0.388).
The case d
≥ 4 uses heavily a recent multiscale renormalization method developed
by Noam Berger. This talk is based on joint works with Alexander Drewitz from
ETH Z
urich.
l

I will introduce a parabolic flow of almost K\"ahler structures,
providing an approach to constructing canonical geometric structures on symplectic manifolds. I will exhibit this flow as one of a family of parabolic flows of almost Hermitian structures, generalizing my previous work on parabolic flows of Hermitian metrics. I will exhibit a long time existence obstruction for solutions to this flow by showing certain smoothing estimates for the curvature and torsion. Finally I will discuss the limiting objects as well as some open problems related to the symplectic
curvature flow.