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.

In this talk I will describe the progress that has been made so far concerning the existene of global strong solutions to the L^{2}-critical defocusing semilinear Schroedinger equation. A long standing conjecture in the area is the existence of a unique global strong L^{2} solution to the equation that in addition scatters to a free solution as time goes to infinity. I will demonstrate the proofs of partial results towards an attempt for a final resolution of this conjecture. I will concentrate on the low dimensions but give the flavor of the results in higher dimensions for general or spherically symmetric initial data in certain Sobolev spaces. Many authors have contributed to the theory of this equation. I will convey my personal involvement to the problem and the results that I have obtained recently. Part of my work is in collaboration with D. De Silva, N. Pavolovic, G. Staffilani, J. Colliander and M. Grillakis.

The modern theory of Dynamical Systems is in major part an offspring of celestial mechanics. Poincare proved non-integrability of the three body problem when he discovered the homoclinic picture. Alexeev explained the existence of the oscillatory motions (a planet approaches infinity but always returns to a bounded domain) in Sitnikov model (one of the restricted versions of the three body problem) using methods of hyperbolic dynamics.
We show that the structures related to the most recent results in the smooth dynamical systems (area preserving Henon family and homoclinic bifurcations, persistent tangencies, splitting of separatrices) also appear in the three body problem. In particular, we prove that in many cases the set of oscillatory motions has a full Hausdorff dimension.

The existence of K\"ahler-Einstein metrics on a compact K\"ahler manifold
of definite or vanishing first Chern class has been the subject of intense
study over the last few decades, following Yau's solution to Calabi's
conjecture. The K\"ahler-Ricci flow is the most canonical way to construct
K\"ahler-Einstein metrics. We define and prove the existence of a family
of new canonical metrics on projective manifolds with semi-ample canonical
bundle, where the first Chern class is semi-definite. Such a generalized
K\"ahler-Einstein metric can be constructed as the singular collapsing
limit by the K\"ahler-Ricci flow on minimal surfaces of Kodaira dimension
one. Some recent results of K\"ahler-Einstein metrics on K\"ahler
manifolds of positive first Chern class will also be discussed.