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.

We will establish a bijective correspondence between finite type associative cones in $\R^7$ and their spectral data, which consists of a hexagonal algebraic curve and a planar flow of line bundles in its Jacobian. We characterize the spectral data by identifying various symmetries on them. We prove generic smoothness of these spectral curves, compute their genus, and compute the dimension of the moduli of such curves. Then we identify a Prym-Tjurin subtorus of the Jacobian, in which the direction of the flow must lie, and compute its dimension. Finally we characterize finite type special Lagrangian cones in $\C^3$ as a subclass of such associative cones in terms of the spectral data. These computations are mainly motivated by Hitchin's recent work on G_2 spectral curves and Langland duality.

In this talk, we establish an analytic foundation for a fully non-linear equation $\frac{\sigma_2}{\sigma_1}=f$ on manifolds with positive scalar curvature. This equation arises from conformal geometry. As application, we prove that, if a compact 3-dimensional manifold $M$ admits a riemannian metric with positive scalar curvature and $\int
\sigma_2\ge 0$, then topologically $M$ is a quotient of sphere.