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.

The high temperature phase of the Sherrington-Kirkpatrick model of spin glasses is solved by the famous Thouless-Anderson-Palmer (TAP) system of equations. The only rigorous proof of the TAP equations, based on the cavity method, is due to Michel Talagrand. The basic premise of the cavity argument is that in the high temperature regime, certain objects known as `local fields' are approximately gaussian in the presence of a `cavity'. In this talk, I will show how to use the classical Stein's method from probability theory to discover that under the usual Gibbs measure with no cavity, the local fields are asymptotically distributed as asymmetric mixtures of pairs of gaussian random variables. An alternative (and seemingly more transparent) proof of the TAP equations automatically drops out of this new result, bypassing the cavity method.

Let alpha be an automorphism of a hyperelliptic curve C of genus g,
and let alpha' be the automorphism of P^1 induced by alpha.
Let n be the order of alpha and let n' be the order of alpha'.
We show that the triple (g,n,n') completely determines the
characteristic polynomial of the automorphism alpha^* of the
Jacobian of C, unless n is even, n=n', and (2g+2)/n is even,
in which case there are two possibilities. We give explicit
formulas for the characteristic polynomial in all cases.