In this lecture, we present a new proof of Perelman's collapsing theorem for 3-manifolds with boundary which is needed for his work on Thurston's Geometrization Conjecture. Among other things, we use an observation of Hamilton-Perelman on incompressible tori boundary for Ricci flow with surgery on thick part of a 3-manifold. Starting from incompressible tori boundary of thin part of 3-manifold, we found that there is an injective F-structure in the sense of Cheeger-Gromov. Consequently, the part of a 3-manifold for Ricci flow with surgery becomes an aspherical graph-manifold, Perelman's collapsing theorem for 3-manifolds follows.

Cayley 4-folds are calibrated (and thus minimal) submanifolds in R^8 associated to a Spin(7) structure. Cayley cones in R^8 that are ruled by oriented 2-planes are equivalent to pseudoholomorphic curves in the grassmanian of oriented 2-planes G(2,8). The twistor fibration G(2,8) -> S^6 is used to prove the existence of immersed higher-genus pseudoholomorphic curves in G(2, 8). These give rise to Cayley cones whose links have complicated topology and that are the asymptotic cones of smooth Cayley 4-folds. There is also a Backlund transformation (albeit a holonomic one) that can be applied globally to pseudo-holomorphic curves of genus g in G(2,8) and this suggests looking for nonholonomic Backlund transformations for other systems that can be applied globally.

We investigate the submanifold geometry of Hermann actions on Riemannian symmetric spaces. After proving that the curvature and shape operators of these orbits commute, we calculate the eigenvalues of the shape operators in terms of the restricted roots of the symmetric space. As an application, we obtain an explicit formula for the volumes of the orbits.

This is joint work with Gudlaugur Thorbergsson.