We attempt to prove positivity of the Lyapunov exponent for
the one-dimensional, discrete, quasi-periodic Schrodinger operator in the
very general case of a smooth, non-transversal (e.g. non-flat at any
point) potential function. This result would hold for all energies. The
method used improves on some techniques developed recently by K. Bjerklov.
These techniques are reminiscent of the ones used to study the dynamics of
the Henon map by M. Benedicks and L. Carleson.

We consider the rate of spread of a body of passive tracers moving under the influence of a random evolving vector field.
The vector field is of a type used as a model for ocean currents and was introduced by Kolmogorov. The rate of growth of the diameter of the body is of interest for practical reasons (such as in problems of pollution control) and we specify its rate of growth.

I will give an overview of the recent work I have done on stochastic modeling of cancer. I will first talk about the concept of multistage carcinogenesis and how we can describe cancer as "bad evolution" within an organism. I will introduce some simple models and explain the phenomenon of "stochastic tunneling". Then I will talk about the role of stem cells in cancer initiation and present some hypotheses about the cellular origins of colon cancer.

Finally, I will talk about growing cellular colonies and models of treatment: how does resistance arise and what can we do about it? Therapies which target specific molecular alterations in cancer cells have shown promising results. Resistance, however, poses a problem, especially in advanced disease. An example is the treatment of chronic myeloid leukemia (CML) blast crisis with Gleevec. I will elucidate the principles which underlie the emergence of drug resistance in cancer. The model (a birth-death process on a combinatorial mutation network) is based on measurable parameters: the turnover rate of tumor cells, and the rate at which resistant mutants are generated. In the context of CML, the prediction is that a combination of three drugs can successfully treat blast crisis.

This is an introduction to the important aspects of
covers of the integers by residue classes and covers of groups
by cosets or subgroups. The field is connected with number theory,
combinatorics, algebra and analysis. It is quite fascinating, and
also very difficult (but the results can be easily understood).
Many problems and conjectures remain open, some nice theorems and
applications will be introduced.