We consider a difference-differential equation
$$\sum_{k=1}^{m}\sum_{j=0}^{q_k}a_{kj}(z)\phi^{(j)}(z+\tau_k)=\gamma(z)\eqno{(*)}$$
where $\tau_1

I will consider the random walk on Z^d driven by a field of random i.i.d.
conductances.
The law of the conductances is bounded from above; no restriction is posed on the
lower tail (at zero) except that the bonds with positive conductances percolate.
The presence of very weak bonds allows the random walk in a finite box mix pretty
much arbitrarily slowly. However, when we focus attention on the return probability
to the starting point -- i.e., the heat-kernel -- it turns out that in dimensions
d=2,3 the
decay is as for the simple random walk. On the other hand, in d>4 the heat- kernel
at time 2n may decay as slowly as o(1/n^2) and in d=4 as slowly as O(n^{-2}log n).
These upper bounds can be matched arbitrarily closely by lower bounds in particular
examples. Despite this, the random walk scales to Brownian motion under the usual
diffusive scaling of space and time. Based on joint works with N. Berger, C. Hoffman,
G. Kozma and T. Prescott.

The talk will discuss the behavior of the spectral measure
at the edge of the spectrum for half line discrete Schroedinger
operators with slowly decaying monotone potentials. A central example
is the bottom of the spectrum for the potential V(n) = 1/n^b, where
0 < b < 1/2. This is joint work with Y. Kreimer and B. Simon.

We discuss the spreading of an initially localized wavepacket
on the Fibonacci chain under Schr\"odinger dynamics. After briefly
recalling the known results that bound the associated dynamical
quantities from above and below, we present a combinatorial approach to
this problem that leads to improved lower bounds (joint work with Mark
Embree and Serguei Tcheremchantsev)