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Description

Our project "SubTile" is focused on the study of non-periodic tilings, in particular of those tilings which are generated by local matching rules or by substitution rules. Perhaps the most famous exemples of such objects are the Penrose tilings. To get a flavour of what to expect, have a look at the tilings encyclopedia. The interest in such tilings has been growing rapidly in recent years, not only because they arose in rather different areas of mathematics and physics (such as logics, number-theory, and quasicrystals) but also because their study employs various different modern mathematical techniques like combinatorics, algebra, dynamical systems and ergodic theory, algebraic topology, non-commutative geometry, statistical mechanics and computer science. The project gathers several recognized specialists in one or more of these fields with notable contributions in the theory of tilings. We have complementary and interrelated goals which include also the creation of a software package to build and manipulate tilings, and the diffusion of our findings towards schools, universities, museums and on websites.

This project includes several research directions:

  • Goodman-Strauss' problem. In one dimension, subshifts of finite type and substitutional subshifts fall in two disjoint classes. In higher dimensions, according to Goodman-Strauss' result, every substitution tiling is a subshift of finite type. However, the relation between the substitution and the local matching rule is at the moment not well understood and not really explicit. The software package should be of a great help here.
  • The Pisot conjecture about the discreteness of the spectrum of Pisot substitutions. We intend to investigate will the spectrum of tiling systems. There is still a lot to do in one dimension, the conjecture has only been solved for substitutions with an alphabet of two letters. Several equivalent forms of this conjecture are known (notably in terms of "coincidence"). Some of the work in one dimension was generalized by Solomyak to higher dimensional substitution tilings. It would be interesting to find another form of coincidence in the higher dimensional Pisot case. We would also like to find a combinatorial version of the conjecture. In particular, we would like to have a purely mechanical way to decide whether a given tiling system has discrete spectrum.
  • Existence and stability of quasi-crystals. Substitution tilings have been widely used to model the atomic structure of quasicrystals. From the combinatorial, ergodic or spectral properties of these tilings, one can deduce certain physical properties of quasi-crystals. To give an example, the spectrum of the dynamical system associated with a tiling provides information on the diffraction spectrum of the quasi-crystal. But we still do not understand really this correspondance: why does the fast cooling of certain alloys produce quasi-crystals?
  • Explicit computation of cohomological invariants. Algorithms are known for the calculation of the cohomological invariants of substitution tilings of small dimension (up to 2) or for canonical cut and project tiling of small codimension (up to codimension 3). The tilings we are interested have fractal acceptance domains, however, something which cannot be handled by the above methods. Hence new ideas are needed.
  • Construction of spectral triples for tilings. Recently, Pearson and Bellissard have proposed a spectral triple for Cantor sets equipped with an ultra metric. This construction can be applied to the discrete hull of a tiling. We would like to understand what these data of non-commutative geometry really say about the tiling. Furthermore we aim to generalize the spectral triple to the whole tiling algebra.
  • Software for visualization of tilings. The interest of a free tiling software package has been explained above. The production of images is essential in this domain to understand the tiling, and also for the diffusion of our results, to other mathematicians, or to students. The members of this project have strong capacities in terms of diffusion of knowledge. We aim to develop and structure these capacities, and share them with the general public. This can be done through conferences, workshops and visit, specially in the partner of this project.

Members

Project coordinator: Pierre Arnoux, IML, Marseille

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Page last modified on October 03, 2009, at 08:33 AM