In this talk I will present some recent results
on perturbations of almost-periodic Jacobi matrices with a
finite number of gaps in the spectrum. In particular, I will
discuss a Szego-type theorem which provides a description of
all Jacobi matrices with spectral measures satisfying a
Szego-type condition. I will also address a limit almost
periodic behavior of coefficients for such Jacobi matrices.

This talk is based on joint work in progress with Jacob
Christiansen and Barry Simon.

I will first briefly review the classical,
"Becker-Doring" (BD) theory of nucleation and describe
the solution of the discrete time-dependent BD equation.
Then, I will discuss low-temperature nucleation in a
two-dimensional Ising system driven by Glauber/Metropolis
dynamics. Here, accurate values of the nucleation rate can
be derived and used to assess the phenomenological BD
picture.

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.