A report by Geertje Hek (Mathematics Department Utrecht University).

A mixed group of about 25 Master Class and Ph.D. students from different countries attended this summer school. Four speakers gave lectures about different topics in spatio-temporal patterns. A brief survey is given here:

Rick Durrett from Cornell gave a view of oriented percolation and contact processes, from the basic theory to recent results. He discussed contact processes with one species, without and with fast stirring (leading to a dynamical system) and ended up with contact processes with two species and fast stirring. With ODE and phase plane theory he derives conditions for coexistence. In an epidemic model and a competing contact process, both with large homogeneous mixed populations, he showed that conditions for invasability and coexistence depend strongly on the ratio of birth rate and death rate.

Nick Ercolani from Arizona told about singularity and defects, patterns far from threshold. By means of examples (Rayleigh-Bénard convection, Swift-Hohenberg) he showed a method to give a good description of patterns with different régimes and a kind of boundary layers. A phase equation was derived, and solvability conditions leaded to a universal `phase gradient form'. Solutions were constructed via Legendre transform and a Hodograph equation. These solutions appeared to give a good description of certain patterns found in experiments.

Yasamasa Nishiura from Sapporo told about patterns in singularly perturbed problems and order parameter equations. By means of examples he gave a new approach to understand singularly perturbed problems (without the knowledge of scaling): the renormalisation method or envelope approach. The knowledge that a solution is given by the envelope of local approximations leads to a renormalisation group equation or map. This method, invariant manifold theory and bifurcation theory gave (stable) patterns which were also found in numerical and real experiments. A video showed us the results.

Satya Majumbar gave a survey of self-organized criticality in sandpile models. A square lattice with piles of particles with certain maximum height formed the model. Adding particles and toppling at one site let particles leave other sites. This gives a matrix that describes the model. One can find stable configurations, recurrent and transient ones. Avoiding any forbidden subconfiguration a class of allowed configurations can be found. The number of allowed configurations appeared to be the determinant of the matrix describing the process.

The lectures were interesting and the group was interested, so there was a lot of discussion. But mathematics were not the only topical items: different cultures, music, the food and world-problems also filled important parts of conversations. The campus of the U.T. served as a place to eat, sleep, walk, go out, sing, etc. One night we however left the campus for pizza and beer in the city of Enschede, where the lecturers turned into coin-football players and great laughers. The mixture of mathematics and joy made this week a very nice one.

Tue Dec 29 15:54:50 MET 1998