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

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 do such a random tiling satisfu 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 ANR-funded project QuasiCool
(2013-2017).

I was the coordinator of the CNRS-funded (PEPS) project Stochasflip (2009-2011).

I have co-organized: