I am basically interested in tilings, namely in the following issues:

  • Compact packings. The problem of finding the densest packing of unit spheres in a given dimension is well studied (e.g., Kepler conjecture in dim 3). If spheres can be of different sizes, much less is known despite evident motivation in material sciences. In particular, does there exist a set of spheres such that the densest packing can be proven to be aperiodic? Promising packings are the compact ones: the graph which connects the centers of any two adjacent spheres of such a packing can be seen as a tiling by simplices, with matching rules to ensure that tiles form a packing.

  • Flip spaces. A flip is the 180° rotation of a hexagon tiled by three rhombi. This is an elementary operation on the set of tilings of a given domain of the plane. The flip space of a domain is the graph whose vertices are the tilings of this domain, with an undirected edge connecting two vertices if the corresponding tilings differ by a flip. What can be said on the structure of this graph? What about the natural higher dimensional generalization?

  • Aperiodic tilings. Robert Berger constructed in 1964 the first aperiodic tiling of the plane, that is, a non-periodic tiling characterized only by local constraints. Lot of work has since been carried out, but there are still many open questions. For example, among the tilings which digitalize irrational subspaces of a given higher dimensional space, can we find an algebraic characterization of those which are aperiodic? How simple can the corresponding local constraints be?

  • Quasicrystals. Quasicrystals were discovered in 1982 (which earned the Nobel Prize to Dan Shechtman in 2011). They are non-periodic material which are however as ordered as crystals.  Aperiodic tilings turned out to provide a good model, with local constraints modeling short-range energetic interactions. In this context, a characterization of aperiodic tilings could be seen as an extension to quasicrystalline structures of the Bravais-Fedorov characterization of crystalline structures. But this does not address the issue of quasicrystal growth...

  • Random tilings. Pick a tiling uniformly at random among all the tilings of a given finite domain. Which properties does such a random tiling satisfy with high probability, that is, with a probability which goes towards one when the size of the domain goes to infinity? In other words, what is the typical look of a random tiling? This problem has been rather deeply investigated for dimer tilings (e.g., by James Propp, Richard Kenyon or Andreï Okounkov), but the case of tilings which naturally appear in the study of aperiodic tilings and quasicrystals remains much more uncharted.

  • I am the coordinator of the 80'Prime CNRS project Predictive self-assembly of supercrystals (2019--2020).
    I was the coordinator of the ANR-funded project QuasiCool (2013-2018).
    I was the coordinator of the CNRS-funded (PEPS) project Stochasflip (2009-2011).
    I have co-organized:
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