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.

Topological field theories have long been expected to be closely related to integrable systems. A famous conjecture of Witten (proven by Kontsevich and others) states that the generating function of descendant integrals on the moduli spaces of curves is a solution to the KdV hierarchy. As a generalization of this result one may speculate a relationship between Gromov-Witten theory and integrable systems. In this talk we give a survey on this conjectural relationship and discuss some (very) low dimensional examples.