I'll begin with the Battelle Rencontre "Lectures in Mathematics and Physics" (Seattle, 1967) and end with S-duality as reflected in Mirror Symmetry and in the Electric-Magnetic Duality connection with the Geometric Langlands Program. In between will be a guided tour of special moments in the interaction of mathematicians and high energy theorists: gauge theory and fibre bundles, instantons and index theory, string theory and Calabi-Yau manifolds.

We discuss some recent developments in the problem of Khler metrics of constant scalar curvature and stability in geometric invariant theory. In particular, we discuss various notions of stability, both finite and infinite-dimensional, and various analytic methods for the problem. These include estimates for energy functionals, density of states and Tian-Yau-Zelditch and Lu asymptotic expansions, geometric heat flows, and both a priori estimates and pluripotential theory for the complex Monge-Ampre equation.

The two-layer beta-plane quasi-geostrophic (QG) model plays a central role in theoretical studies of atmospheric and oceanic dynamics. It is a pair of coupled non-linear partial differential equations involving functions of two space variables and one time variable (streamfunctions for coupled two-dimensional flows). Solutions represent flows in a sense intermediate between 2-d and 3-d flows: they have a mild form of the ``vortex stretching'' process, absent in 2-d flows, that is at the heart of the difficulty in proving the long-time existence of classical solutions to the
3-d Navier-Stokes equations.

Numerical solutions to these QG equations display analogues of important features of atmospheric and oceanic flow, some of which I will illustrate. As is true of climate models, many interesting features are revealed only by long time averaging of the numerical solutions. The results I will present, on long-time existence of regular solutions and on dissipativity,
are part of an effort to provide a rigorous justification for this averaging, something beyond our reach in the case of the vastly more complicated climate models.

The talk will place the model in the context of other QG models, point out a useful formal similarity to the Kuramoto-Sivishinsky equation, and sketch proofs of the main results. The work is joint with C. Foias, C. Onica, E. Titi, and M. Ziane.

Let $X^n\subset \Bbb C\Bbb P^{n+1}$ be a hypersurface defined as the zero set of a degree $d$ polynomial with $d\leq n$. Such hypersurfaces have lines through each point $x\in X$. Let $\mathcal C_x\subset \Bbb P(T_xX)$ denote the set of tangent directions to lines on $X$ passing through $x$. Jun-Muk Hwang asked how $\mathcal C_x$ varies as one varies $x$. The answer turns out to be interesting, with two natural exterior differential systems governing the motion. In addition to describing these EDS and some immediate consequences, I will also discuss applications to questions in computational complexity and algebraic geometry. This is joint work with C. Robles.

Elliptic divisibility sequences arise as sequences of
denominators of the integer multiples of a rational point on an elliptic
curve. Silverman proved that almost every term of such a sequence has a
primitive divisor (i.e. a prime divisor that has not appeared as a
divisor of earlier terms in the sequence). If the elliptic curve has
complex multiplication, then we show how the endomorphism ring can be
used to index a similar sequence and we prove that this sequence also
has primitive divisors. The original proof fails in this context and
will be replaced by an inclusion-exclusion argument and sharper
diophantine estimates.