Let $K$ be a number field and $E/K$ an elliptic curve without
complex multiplication. A well-known theorem of Serre asserts that the
Galois group of $K(E[\ell])/K$ is as all of ${\rm GL}_2(\Z/\ell)$ for any
sufficiently large prime $\ell$. If we replace $E/K$ by a polarized abelian
variety $A/K$ with trivial endomorphism ring, then Serre later showed
that the Galois group of $K(A[\ell])/K$ is also as large as possible, for
all sufficiently large $\ell$, provided $\dim(A)$ is 2,6 or odd. We will
show how to prove a similar result for `most' $A$ and without any
restriction on $\dim(A)$.

Magnetic Resonance Imaging (MRI) uses the resonance of the nucleus of
chemical species to generate images of their spatial distribution. In
medical MRI, a simple model considers tissue as made up of only water
and fat. Most of the clinically relevant information is in the water
signal and the fat signal is considered clutter to be suppressed. The
separation of water and fat based on the difference of their resonance
frequencies using multiple images provides a robust method for fat
suppression in areas where the magnetic field is inhomogeneous. In this
talk, we will show how to propagate the uncertainty due to imperfections
of the magnetic field into our estimate of water and fat. The
optimization of data acquisition based on the Cramer-Rao bound (CRB) for
this nonlinear problem leads to new optimal solutions which do not arise
when the magnetic field is assumed to be homogeneous. A reconstruction
based on maximum likelihood estimation allows us to achieve the CRB for
realistic noise levels which is verified by Monte Carlo simulations,
experimentally and clinically. Our acquisition and reconstruction is
part of a method for chemical species separation currently used by
General Electric MRI scanners.

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.