A system of non-linear partial differential equations modeling tumor invasion into surrounding healthy tissue is analyzed. The model focuses on key components involved in tumor cell migration and takes into account cell motility and haptotaxis. The latter means the directed migratory response of tumor cells to the extracellular environment. Individual cell processes are modeled according to cell age. The equation for the tumor cell density thus incorporates second-order terms representing diffusion and taxis as well as a first-order part due to cell aging. Global existence and uniqueness of nonnegative solutions is shown.

I will be talking about three different pedagogical activities at Harvard in which I'm involved:

* An undergraduate course in multivariable mathematics for social sciences
* An online placement exam web application
* The A.L.M in Mathematics for Teaching program, serving area middle- and high-school teachers

Often new teachers and tutors are given extensive training on general ideas and principles of good teaching. There may be little or no link between these ideas and the logistics of how to implement them within the courses they will be teaching/assisting.

In teaching a recent course for Quantitative Learning Tutors at the University of Connecticut, I sought to design a curriculum which closely ties good teaching/tutoring practices with specific science course content. I will present the learning goals for this course, specific
examples of projects and activities, and student learning assessment. This course contained a significant online component which will be outlined.Finally, I will describe how this curriculum can be applied to TA training
in mathematics.

A number of recent experiments have shown that several organisms
that reproduce by fissioning (e.g. E. coli bacteria)
don't share the cellular damage they have
accumulated during their lifetime equally among their offspring. Using
a stochastic PDE model, David Steinsaltz and I have shown that under quite
general conditions the optimal asymptotic growth rate for a population
of fissioning organisms is obtained when there is a non-zero but moderate
amount of preferential segregation of damage -- too much or too little
asymmetry is counter-productive. The proof uses some new results of ours
on quasi-stationary distributions of one-dimensional diffusions and
some Sturm-Liouville theory. The talk is intended for a probability
audience and I won't assume any knowledge of biology.