I'll begin by describing our recent attempts to model the evolution of crypts in the colon, using methylation patterns as markers [see Yatabe et al.]. Mutations in colon crypts are thought to play an important role in pre-tumor progression, and therefore in understanding the time to cancer. One common complaint about such multistage and multihit models is that they require unrealistically high mutation rates to explain the observed incidence of cancer. I'll use our model, together with classical extreme value theory, to show that we can explain the SEER incidence data for colon cancer using typical mutation rates [Calabrese et al.]. A number of corroborative datasets and open problems will be discussed.

In this talk, we will show that the a.c. spectrum of the multidimensional Dirac operator is being preserved under very weak perturbations. The conditions on the decay of potential are optimal in some sense. The case
of Schrodinger operator will be discussed too.

We define and study spaces of random operators which satisfy Hilbert-Schmidt and trace class - like conditions with respect to the trace per unit volume.

Such spaces appear in the derivation of the Kubo formula in linear response theory for random Schrodinger operators in a constant magnetic field.

We study the (signed) flow of spectral multiplicity for a family of magnetic Schrodinger operators in R^2,
$$
H(\lambda a) = (-i \nabla -\lambda a)^2 +V(x),
\lambda \ge 0,
$$
in the large coupling limit, i.e., $\lambda \to 0$. Our main assumption is for the magnetic field
$ B curl \lambda a$ to have compact support consisting of a finite number of components. The total magnetic flux may be non-zero.