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.

The development of quantum mechanics in the 1920s convinced physicists that future progress in understanding solids would flow from that discipline, and so continuum mechanics of solids was abandoned by physicists. Thus, success of the latter in late twentieth century is regarded as surprising by most.

We present an overview of the construction of a very general Lagrangian of a closed system of a dielectric crystal interacting with the electromagnetic field. The Lagrangian is first constructed for discrete particles, a long-wavelength (continuum) limit is taken in a manner to preserve all of the eigenmodes. The crystal can be of any class of symmetry, have any structural complexity, and have interactions between its various eigenmodes and between them and the electromagnetic field to any order of nonlinearity. All eigenmodes are included: electromagnetic, acoustic and optic modes of vibration, spin, and all polaritonic combinations of them.

The photoelastic effect was show to have been wrongly formulated for 155 years: the independent variable characterizing the deformation had been wrong! Thus, the interaction tensor has a more general symmetry. The accepted relation between the photoelastic effect and electrostriction was shown to have been wrong for almost as long. The elastic stiffness tensor was shown to lose its traditional symmetry when a soft optic mode became involved. All treatments of acoustic harmonic generation in piezoelectrics were shown to be wrong. The best derivation of optical activity was shown to have missed a fundamental contribution having a different dispersion. The Abraham - Minkowski controversy about the momentum of a light wave in a medium was resolved. The most general Poynting vector in a medium was found. Several nonlinear interactions were characterized and interpreted for the first time.