Department of Mathematics

I will begin by explaining the theory of calibrations and present some important examples. These examples are cases where one has large families of minimal (in fact, homologically volume minimizing) varieties. The basic case is K\”ahler geometry, where the varieties are the complex analytic ones. Now for a general calibrated manifold there is no analogue of the holomorphic functions. However, it turns out that there is always a complete analogue of the plurisubharmonic functions (those which are subharmonic on complex curves), and many of the results from several complex variables carry over. These functions are, in a sense, dual to the subvarieties. Furthermore, the inequality defining general plurisubharmonic functions also gives an equality, that is, a nonlinear partial differential equation. These associated equations are generalizations of the complex Monge-Amp\`ere equation in the K\”ahler case. We now have a good understanding of how to solve the Dirichlet Problem for solutions to these equations. An example some people may like concerns symplectic manifolds equipped with a Gromov metric. The associated plurisubharmonic functions are those whose Hessian has non-negative trace on every Lagrangian plane. The corresponding differential operator is a polynomial in tr$(D^2 u)$ and the skew-Hermitian part of $D^2 u$. Tea at 3:30 pm in Stevenson 1425. (Contact Person: Ioana Suvaina)