We discuss d+1 dimensional percolation models with d dimensional
quasiperiodic disorder. A multiscale scheme is introduced which is suited
to the spatial structure of quasiperiodic disorder. In this case we will
show almost sure stretched exponential decay of correlations as compared
to faster than polynomial decay of correlations obtained for similar
models with random disorder. We mention in this case a disorder-rated
transition of phase structure.

Our main question is the p-adic meromorphic continuation of
the L-function attached to a p-adic character for the rational
function field over a finite field of characteristic p. In this talk,
I will explain a new and (hopefully) transparent approach to this
problem. (This is ongoing joint work with Chris Davis).

The Perona-Malik equation is a celebrated example of nonlinear forward-backward diffusion, introduced in the context of image denoising. It can be viewed as the formal gradient-flow of a functional with a convex-concave integrand. In spite of its mathematical ill-posedness, numerical experiments exhibit better than expected behavior of its solutions.
After a general introduction to the equation itself, we present a few approximation schemes, some classical and some more recent. The approximating solutions show distinct behavior on three time scales (called fast, standard, and slow time). We provide a rigorous explanation for the slow time behavior of the different approximations.
In order to carry out this analysis, we prove an abstract result about passing to the limit in gradient-flows (in the more general context of the theory of maximal slope curves in metric spaces). We are guided by the general principle that "the limit of a family of gradient-flows is the gradient-flow of the limiting functional".