Whether the 3D incompressible Euler or Navier-Stokes equations
can develop a finite time singularity from smooth initial data has been
an outstanding open problem. Here we review some existing computational
and theoretical work on possible finite blow-up of the 3D Euler equations.
We show that the local geometric properties of vortex filaments can lead
to dynamic depletion of vortex stretching, thus avoid finite time blowup
of the 3D Euler equations. Further, we perform large scale computations of
the 3D Euler equations to re-examine the two slightly perturbed anti-parallel
vortex tubes which is considered as one of the most attractive candidates
for finite time blowup of the 3D Euler equations. We found that there is
tremendous dynamic depletion of vortex stretching and the maximum vorticity
does not grow faster than double exponential in time. Finally, we present
a new class of solutions for the 3D Euler and Navier-Stokes equations,
which exhibit very interesting dynamic growth property. By exploiting
the special nonlinear structure of the equations, we can prove the global
regularity of this class of solutions.

The use of antiviral drugs has been recognized as the primary public
health strategy for mitigating the severity of a new influenza pandemic
strain. However, the success of this strategy requires the prompt onset of
therapy within 48 hours of the appearance of clinical symptoms. The
evolution of drug resistance in the virus can also pose a problem for
antiviral treatment strategies. I'll present a compartmental model that
monitors the density of infected individuals in terms of the time elapsed
since the onset of symptoms. Such a model can be expressed by a system of
delay differential equations with both discrete and distributed delays,
and is based on the interaction between viral dynamics at the host level
and the spread of the disease in the population. It shows that treatment
alone is unlikely to control an outbreak unless other control measures to
reduce the spread of disease are also in place. Furthermore, we show that
levels of treatment that have a chance of controlling the disease will
also drive the emergence of drug resistant outbreaks. While an antiviral
treatment is helpful for containing a pandemic, its effectiveness depends
critically on timely and strategic use of drugs.

I will give a survey of what is known at present concerning the way solutions to the Porous medium equation "fill holes" in their support. There turn out to be some surprising similarities with Euclidean and Affine Curve Shortening.