The level set method is a numerical technique to track moving interfaces. I will discuss my research on a finite element based level set method and present some simulations of crytal growth and dendritic solidification.

The study of gene orders for constructing phylogenetic trees was
introduced by Dobzhansky and Sturtevant in 1938. Different genomes
may have homologous genes arranged in different orders. In the early
1990s, Sankoff and colleagues modelled this as ordinary (unsigned)
permutations on a set of numbered genes $1,2,\ldots,n$, with
biological events such as inversions modelled as operations on the
permutations. Signed permutations are used to indicate the relative
strands of the genes, and circular permutations may be used for
circular genomes. We use combinatorial methods (generating functions,
asymptotics, and enumeration formulas) to study the distributions of
the number and lengths of conserved segments of genes between multiple
genomes, including signed and unsigned genomes, and circular and
linear genomes. This generalizes classical work from the 1940s--60s
by Wolfowitz, Kaplansky, Riordan, Abramson, and Moser, who studied
decompositions of permutations into strips of ascending or descending
consecutive numbers. In our setting, their work corresponds to
comparison of two unsigned genomes (known gene orders, unknown gene
orientations).

We study impulse control problems of jump diffusions with delayed
reaction. This means that there is a delay $\delta>0$ between the
time when a decision for intervention is taken and the time when the
intervention is actually carried out.
We show that under certain conditions this problem can be transformed
into a sequence of iterated no-delay optimal stopping problems and
there is an explicit relation between the solutions of these two
problems.
The results are illustrated by an example where the problem is to
find the optimal times to increase the production capacity of a firm,
assuming that there are transaction costs with each new order and the
increase takes place $\delta$ time units after the (irreversible)
order has been placed.

The presentation is based on joint work with Agn\`es Sulem: ``Optimal
stochastic control with delayed reaction", Applied Mathematics and
Optimization (to appear)

The purpose of this lecture is to give a non-technical, yet rigorous introduction to Malliavin calculus for L$\acute{e}$vy processes and its applications to finance. The lecture consists of two parts:

Part 1 deals with the Brownian motion case. We first use the Wiener-It\^{o} chaos expansion theorem to define the Mallavin derivative in this context and then study some of its fundamental properties, including the chain rule and the duality property (integration by parts). Then we apply it to finance. Examples of applications are

(i) the hedging formula in complete markets provided by the Clark-Ocone theorem,

(ii) ``parameter sensitivity results", e.g. a numerically tractable
computation of the ``delta-hedge" and other``greeks" in finance.

Part 2 deals with the general L$\acute{e}$vy process case. To some extent a similar presentation of the Malliavin derivative can be given here as in Part 1, but there are also basic differences, for example regarding the chain rule. Examples of applications to finance are

(i) optimal hedging in incomplete markets (based on the Clark-Ocone formula L$\acute{e}$vy processes),

(ii) optimal consumption and portfolio with partial information in a market driven by L$\acute{e}$vy processes.

The presentation is mainly based on the forthcoming book
G. Di Nunno, B. $\O$ksendal and F. Proske:
``Malliavin Calculus for L$\acute{e}$vy Processes and Applications to Finance".
Springer 2008/2009 (to appear).