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.

In many applications, we are faced with the problem of solving
an ODE with multiple initial conditions. Standard ODE integrators compute
the solution for each initial condition independently, which can be
computationally expensive. The phase flow method (PFM) is a novel approach
to construct phase maps for nonlinear autonomous ordinary differential
equations on their compact invariant manifolds. It first constructs the
phase map for a small time using a standard ODE integrator and then
bootstraps the process with the help of a local interpolation scheme and
the group property of the phase flow. This construction usually takes the
time of tracing a couple of solutions and the resulting approximation to
the phase map is accurate. Once the phase map is available, integrating
the ODE for an initial condition on the invariant manifold only utilizes
local interpolation, thus having constant complexity. We present the
method and prove its properties in a general setting. As an example, the
phase flow method is applied to the fields of high frequency wave
propagation and computational geometry. In high frequency wave
propagation, we concentrate on three problems: wavefront construction,
multiple arrival time and amplitude computation. We also discuss the
adaptive issues in the implementation. In computational geometry, we apply
the phase flow method to generate geodesic flow on smooth 2D surfaces.
Numerical results will be presented as well. (Joint work with Emmanuel
Candes)