We consider sufficient conditions for regularity of weak solutions of the Navier-Stokes equation. By a result of Neustupa and Panel, the weak solutions are regular provided a single component of the velocity is bounded. In this talk we will survey existing and present new results on one component and one direction regularity.

We consider collapsing sequences of solutions to the Ricci flow on 3-manifolds with almost nonnegative curvature. Such sequences may arise from dilating about infinite time singularities. For finite time singularities no collapse can occur by a result Perelman. Using Fukaya theory, we study some geometric (not topological) aspects of such collapse. When the limit solution is 1-dimensional we construct a virtual 2-dimensional rotationally symmetric limit. This is joint work with David Glickenstein and Peng Lu.