I shall review recent results about KAM for space-multidimensional
nonlinear PDE and discuss the two schemes, used now to prove them

We study the self-adjoint Schr\"odinger operator on the axis
\[
H_v = -\frac{d^2}{dx^2} + v(x),\ \ \ -\infty < x < \infty,
\]
with an almost periodic real-valued potential $v(x)$.
Let $\Lambda$ be a dense subgroup of the group $(\R,+)$. Denote by
$AP_{\Lambda}(\mathbf{R})$ the Banach space of all real-valued almost periodic
functions on $\R$ whose all frequencies belong to $\Lambda$, with the supremum norm.
\bigskip

\textbf{Theorem}
\ There exists a dense $G_{\delta}$ subset $X\subseteq AP_{\Lambda}(\mathbf{R})$,
such that for all $v\in X$ the operator $H_v$ has a nowhere dense spectrum.

Self-propagating reaction fronts occur in many chemical and
physicalsystems possessing two key ingredients: a reactive medium (for example a fuel-air mixture in the case of flames) and an autocatalystthat is a product of the reaction which also accelerates the reaction(for example thermal energy in the case of flames). Self-propagation occurs when the autocatalyst diffuses into the reactive medium,initiating reaction and creating more autocatalyst. This enables reaction-diffusion fronts to propagate at steady rates far from anyinitiation site. In addition to flames, propagating fronts have been observed in aqueous reactions, free-radical initiated polymerizationprocesses and even propagating fronts of motile bacteria such as E.coli.

This talk will focus on a comparison of the dynamics of these
four different types of fronts including propagation rates, extinction conditions and instability mechanisms. Our research has shown that despite the disparate nature of the reactants
and autocatalysts in these four systems, remarkably similar dynamical behavior is observed since the underlying driving mechanisms for propagation are similar.The key role of loss mechanisms (heat, chemical species or cell death) and
differential diffusion of reactant and autocatalyst("Lewis number") is demonstrated.

The aim of this talk is to introduce a flow/flood simulation system
which is designed to simulate coupled surface and subsurface flow
problems arising from flood prediction and control. The mathematical
and computational challenges are: (1) model deveoplement and model
reduction, (2) numerical approximation for shallow water equations
with complex physics, (3) algorithm design, (4) computer
simulation/implementation, and (5) validation. The first four
challenges shall be discussed in this talk, and particularly emphasis
will be given to the model development and algorithm design. The talk
should be accessible to general audience in applied/computational
mathematics and hydraulogy.