We call a metric quasi-Einstein if the m-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several
rigidity results. We also give a splitting theorem for some Kahler quasi-Einstein metrics.

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.