Gaven J. Martin (The University of Auckland, New Zealand)
1. The Beltrami equation in the degenerate case.
The existence theorem for quasiconformal mappings
(sometimes called the Generalised Riemann Mapping Theorem) has found
a central role in a diverse variety of areas such as holomorphic
dynamics, Teichm\"uller theory, low dimensional topology and
geometry, and the planar theory of PDEs. Anticipating the needs of
future researchers we give an account of the
``state of the art'' as it pertains to this theorem, that is to the
existence and uniqueness theory of the planar Beltrami equation, and
various properties of the solutions to this equation. Modern
researchers are most interested in the degenerate case where
significant new applications lie and where recent advances in
harmonic analysis have led to an understanding of this equation in
these important settings.
2. Extremal Mappings of finite distortion.
I will present recent joint work with K. Astala, T. Iwaniec and J.
Onninen where we refine the connections between the theory of
mappings of finite distortion and the calculus of variation.
Our primary aim is to extend the study of extremal quasiconformal mappings
by considering integral averages of the distortion function instead of its
L-infinity norm. We identify many new and unexpected phenomena concerning
existence, uniqueness and regularity for the extremal problems. The
principal advantage of minimizing over the family of homeomorphisms in the
above variational problems lies in the fact that the inverse maps are also
extremal for their own energy integrals. Sometimes this associated problem
for the inverse mapping is easier to solve than the original one as it may
involve minimizing a convex functional.
There are many natural reasons for studying such problems. We
eventually hope to lay down the analytical foundations for approaches to
compactifying the moduli spaces, such as Teichmuller spaces, where it is
our expectation that a compactification will be by mappings whose
distortion function lies in some natural integrability class.