Tiling planforms dominated by diamonds (such as the diamond-shaped seeds on a sunflower head), hexagons, or ridges (such as those on saguaro cacti) are observed on many plants. We analyze PDE models for the formation of these patterns that incorporate the effects of growth and biophysical and biochemical mechanisms. The aim is to understand both the underlying symmetries and the information specific to the mechanisms. The patterns are compared to Voronoi tessellations, and we will start to draw a bigger picture of growth and symmetry in biological systems.

The subject of enumerative geometry goes back at least to the middle
of the 19th centuary. It deals with questions of enumerating geometric
objects, e.g.
(a) how many lines pass through 2 points or
through 1 point and 2 lines in 3-space?
(b) how many conics in 2-space are tangent to k lines and
pass through 5-k points?

There has been an explosition of activity in this field over the past
twenty years, following the development of Gromov-Witten invariants in
sympletic topology and string theory. The idea of counting parameterizations
of curves in order to count curves themselves has led to solutions of
whole sets of long-standing classical problems. At the same time, string
theory has generated a multitude of predictions for the structure of
GW-invariants, as well as for the behavior of certain natural families
of Laplacians. It has in particular suggested that there is a diality
between certain symplectic and complex manifolds and that in some cases
GW-invariants see some geometric objects, that are yet to be fully
discovered mathematically.

In this talk I hope to give an indication of what enumerative geometry
is about and of the shift in the paradigm that has occured over the past
two decades.

In this talk I will give a survey of Monte Carlo methods for solving linear systems, and present results of recent research on a new estimator. I will also present numerical results comparing a sequential Monte Carlo method with the standard deterministic solvers. I will then describe quasi-Monte Carlo methods, and present results of current research made on a new modification of the Halton sequence.

Broadband, coherent array imaging can be made quite robust in random media by using interferometric
algorithms that tend to minimize the effect of random inhomogeneities. I will introduce and describe these algorithms in detail, and I will
show the results of several numerical simulations that assess their effectiveness.