Abstract: Geometric flows have been considered by several researchers as an approach to constructing solutions of the special holonomy equations. In this talk, I will examine the nature of these `flows', in particular, their well-posedness and stability. The first nontrivial case, namely SU(2) holonomy in dimension 4, already displays many of the interesting features that are encountered in the exceptional cases (G2 holonomy in dimension 7 and Spin(7) holonomy in dimension 8). The study of these features raises some interesting problems in differential geometry even in dimension 3, as will be explained in the lecture.

Abstract: A considerable amount of work has been done to classify Riemannian submersions and foliations from space forms, and most of the recent work has concentrated on using non-negativity of the curvature to prove global rigidity results. Meanwhile, the local nature of the PDE that define Riemannian submersions has received considerably less attention and, in fact, little appears to be known about the local generality of solutions of the underlying overdetermined PDE. In this talk, I will discuss the nature of this overdetermined PDE system, the cases in which the full local solution space is understood and the prospects for understanding the general case. If there is time, I will also discuss what this information could tell us about the global problem of classifying the Riemannian submersions from space forms.