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.

Initial results from new calculations of interacting anti-parallel Euler vortices are presented with the objective of understanding the origins of singular scaling presented by Kerr (1993) and the lack thereof by Hou and Li (2006). Core profiles designed to reproduce the two results are presented, new more robust
analysis is proposed, and new criteria for when calculations should be terminated are given. Most of the analysis is on a $512\times 128 \times 2048$ mesh, with new analysis on a just completed $1024\times 256\times 2048$ used to confirm trends. The qualitative conclusions of Kerr (1993) are supported, but most of the proposed scaling laws will have to be modified. Assume enstrophy growth like $\Omega\sim (T_c-t)^{-\gamma_\Omega}$ and vorticity growth like $||\omega||_\infty \sim (T_c-t)^{-\gamma}$. Present results would support $\gamma_\Omega\rightarrow 1/4-1/2$ and $\gamma>$. The results are not conclusive since they require higher resolution calculations (work in progress) to further confirm the trends.

Let $B = (B_t: t\ge 0)$ be a real-valued Brownian motion and let
$L = (L_t: t\ge 0)$ denote its local time in state 0. We present a characterization of the measurable functions $H$ such that $M_t = H(B_t,L_t)$
is a continuous local martingale. It turns out that the class of such functions is considerably wider when one relaxes the smoothness conditions that would be needed for a facile application of It\^o's formula.