Let $M$ be a finite dimensional complex manifold. Its loop space
$LM$ is an infinite dimensional complex manifold consisting of
maps (loops) $S^1 \to M$ in some fixed $C^k$ or Sobolev $W^{k,p}$
space. It is a natural question to solve d-bar equation
and/or compute the Dolbeault cohomology groups on loop spaces. I
will talk about the joint work with L. Lempert to identify the
first Dolbeault cohomology group of the loop space of the Riemann
sphere. There are serious obstructions to applying techniques in
finite dimensions to an infinite dimensional setting. We introduced
a bag of tools to overcome the difficulties.

We present a new adaptive numerical scheme for solving parabolic PDEs in
cartesian geometry. Applying a finite volume discretization with explicit
time integration, both of second order, we employ a fully adaptive
multiresolution scheme to represent the solution on locally refined nested
grids. The fluxes are evaluated on the adaptive grid. A dynamical adaption
strategy to advance the grid in time and to follow the time evolution of
the solution directly explaoits the multiresolution representation.
Applying this new method to several test probelms in one, two and three
space dimensions, like convection-diffucion, viscous Burgers and
reaction-diffusion equations, we show its second order accuracy and
demonstrate its computational efficiency.
This work is joint work with Olivier Roussel.