March 3 - 2000:
On March 3 2000, the Mark Kac seminar
will meet again.
11:15-13:00: Michael Baake (Tübingen):
Mathematical Aspects of Aperiodic Order, Part 2.
14:15-16:00: Anton Wakolbinger (Frankfurt):
Voter model and branching particle systems in catalytic media.
The meeting will take place in room 220 of the James Boswell Institute, Bijlhouwerstraat 6, Utrecht.
ABSTRACT MICHAEL BAAKE
Michael Baake is the main speaker of the Mark Kac seminar this year, and he will give three lectures in February, March and April. If you have missed the first lecture, but are interested to come in March, please see the schedule, where there is a link to a survey paper that gives an introduction to quasicrystals.
The discovery of quasicrystals in the early eighties has triggered an intensive investigation of the various kinds of ordered states that are possible between periodic and random. In this series of lectures, I plan to describe the impact that quasicrystals had, focusing specifically of aspects of diffraction theory.
In the first lecture, I will start with the history of the field and summarize how and why quasicrystals challenged our understanding of the solid state. Some generalizations of crystallographic tools will be described that are suitable to cope with symmetry and equivalence concepts in this more general situation.
In the second lecture, I plan to survey mathematical diffraction theory, with special emphasis on the perfectly ordered systems. Of particular interest are point sets (representing atomic positions) which lead to pure point diffraction spectra, whose classification is far from complete.
In the third lecture, I will approach the diffraction of stochastic structures (such as random tilings) with methods from statistical physics. These systems show a variety of different spectral properties, including the possibility of practically relevant examples with singular continuous spectra.
ABSTRACT ANTON WAKOLBINGER
The voter model on the square lattice exhibits a ``diffusive clustering'' - this is a beautiful result of Cox and Griffeath from the mid eighties, and we will review some of its features at the beginning of the talk.
What about if the voting, i.e. the adoption of a neighbour's type, takes place not everywhere on the suare lattice like in the classical voter model, but only in the presence of a catalyst, whereas in catalyst free regions there is a simple exchange of types of neighbours at constant rate? We study this model for the case in which the (space-time dependent) random catalyst itself is a voter model configuration. In the case of two types, and for an ergodic initial configuration, the reactant system converges locally to a mixed Bernoulli configuration whose parameter distribution we can compute and interprete.
Another part of the talk (which as well is based on joint results with Andreas Greven and Achim Klenke) will deal with catalytic branching particle systems. Here, the main tool is a superprocess rescaling of the system.
Time permitting, we will finally touch upon recent results of Cox et al. on superprocess limits of rescaled voter models, thus giving a bridge between the voting and the branching world.