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. I'm particular, does 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
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 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
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.,
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
I was the coordinator of the CNRS-funded (PEPS) project Stochasflip (2009-2011).
I have co-organized: