Aims and Scope
The main theme of the school is the mathematical analysis of probabilistic models of origins and stability of non-periodic structures. This includes models of quasicrystals, probabilistic cellular automata, spin glasses and possible connections between them.
One of the fundamental problems of physics is to understand why particle configurations with the minimal interaction energy should form periodic crystals or exhibit at least some sort of long-range order of non-periodic quasicrystals. One of the main questions addressed during the school is as follows:
Does there exist a translation-invariant finite-range interaction on a regular lattice which would have a non-periodic Gibbs measure at all sufficiently low temperatures? We will also explore connections between non-periodic Gibbs measures and stationary states of probabilistic cellular automata. There are already some new ideas in this direction.
Spin glasses (the other theme of the school) can be modelled as spin systems with random interactions (as opposed to deterministic interactions of quasicrystals). While analogous systems with random external fields are relatively well understood, fundamental questions about short-range spin glass models remain unanswered, including the number of ground states, even in two dimensions. Even less is known about the number and properties of spin glasses in higher dimensional models, including the physically most important dimension three. New techniques are needed to study these problems and the existing methods (metastates, analysis of fluctuations) need to be refined.
The aim of the school is two-fold. First of all we plan to gather in Będlewo leading international experts working on the problems to be addressed in the school. This will make possible to exchange views and ideas on specific well-defined problems of rigorous statistical mechanics of non-periodic systems. We also hope that participants will start new collaborations, establishing working groups. We would also like to attract young people to this area of research. This can be possibly achieved by offering them work projects. We hope to be able to organize a meeting of a working group in the following year.