Any complex harmonic function of finite Dirichlet energy on a Jordan domain has boundary values on the Jordan curve in a sense due to Osborn. For which Jordan curves must there be a harmonic function of finite Dirichlet energy on the complement with these same boundary values?

The Plemelj-Sokhotski jump formula says that a reasonably regular complex function on a reasonably regular Jordan curve can be written as the difference of boundary values of holomorphic functions on the domain and its complement. For which Jordan curves does the jump formula hold in the Dirichlet space setting?

The answer to both of these questions (once they are made suitably precise) is: for those Jordan curves which are quasicircles.

Speaker's web page: http://server.math.umanitoba.ca/~schippers/

Abstract: The main points of this talk are as follows:

- constructions of vector-valued automorphic forms on bounded symmetric domains via Poincare series

- vector-valued automorphic forms associated to submanifolds of the complex unit ball - studying the behavior of asymptotics of the inner product of two Poincare series associated to submanifolds of the complex unit ball, for large weights, with examples.

We formulate a conjecture, analogous to the Hardy--Littlewood prime tuples conjecture, for tuples of sums of two squares. Assuming this conjecture holds, we show that sums of two squares are distributed among the natural numbers as if according to a Poisson process. We discuss numerical evidence, and unconditional results, in support of our conjecture. This is joint work with Pär Kurlberg and Lior Rosenzweig.