April 7- 2000:
On April 7 2000, the Mark Kac seminar will meet again.
11:15-13:00: Michael Baake (Tübingen):
Mathematical Aspects of Aperiodic Order, Part 3.
14:15-16:00: Frank Redig (Leuven):
Abelian sandpiles on infinite lattices.
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 are interested in the subvject, please see below for links to a relevant papers by Michael.
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.
Some files containing extra information on quasycristals:
Review paper:
Two other recent papers:
ABSTRACT FRANK REDIG
In recent years the concept of self-organised criticality (SOC) as introduced in the famous paper of
Bak, Tang and Wiesefeld, has generated much interest. The simplest and most popular model is the so-called
abelian sandpile (AS). In this model particles are added at randomly chosen sites of a (finite) graph. According to certain
``toppling rules" the added particles are redistributed, and can leave the system at the boundaries. The influence of adding a particle
at one site can reach very far (``avalanches are like critical percolation clusters"). This non-locality
of the dynamics is responsible for slowly (as a power) decaying correlations in the
stationary state. Though for the AS many exact results were obtained in the physics
literature, the question of the thermodynamic limit has remained open. We ask and (partially) answer two basic questions:
1) Do the stationary measures of finite volume AS have a (unique) thermodynamic limit ?
2) Is it possible to define a thermodynamic limit of the
finite volume AS dynamics, i.e. a (stationary) Markov process on infinite volume
AS configurations ?