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Tiling and Recurrence (1721)
Pavages et récurrence

Dates: 4-8 December 2017 at CIRM (Marseille Luminy, France)

In this conference, we are interested in the recent developments around the mathematical theory of tilings and its recurrence properties, which have a lot of connections with other areas like number theory, dynamical system, quasi-crystal, computer science and discrete geometry. We intend to focus particularly on the following areas:
Recurrence properties of tiling and number theory
There has been a series of recent developments on bounded remainder sets involving various methods (dynamical, topological, number theoretical, see [2, 3]). We plan to discuss and compare different approaches involved there. We would like to consider frequencies and recurrence properties in tiling spaces, by focusing on variants of ergodic averages in this framework.

Spectral property of tiling dynamical systems
Related to the first theme, we shall discuss the long standing Pisot substitution conjecture, the main remaining problem in this area. Pure discreteness of tiling is essentially the strongest recurrence property we can have, which is equivalent to almost periodicity of the associated point set (c.f. [6, 1]). On this occasion we wish to merge people working in these areas to produce possible breakthroughs.

Aperiodic tile set and quasi-crystals
An aperiodic hexagonal monotile was found by Taylor-Socolar [5]. More recently the number of aperiodic tile set of Wang tiles reached its theoretical minimum 11 with Jeandel-Rao [4]. Their recurrence properties are quite fascinating and we shall discuss them as models of quasi-crystals.
[1] S. Akiyama and J.-Y. Lee, Algorithm for determining pure pointedness of self-affine tilings, Adv. Math. 226 (2011), no. 4, 2855-2883.
[2] A. Haynes, M. Kelly, and B. Weiss, Equivalence relations on separated nets arising from linear toral flows, Proc. Lond. Math. Soc. (3) 109 (2014), no. 5, 1203-1228.
[3] A. Haynes, H. Koivusalo, L.Sadun, and J. Walton, Gaps problems and frequencies of patches in cut and project sets, ArXiv:1411.0578.
[4] E. Jeandel and M. Rao, An aperiodic set of 11 wang tiles, ArXiv:1506:-6492.
[5] J. E. S. Socolar and J. M. Taylor, An aperiodic hexagonal tile, Journal of Combinatorial Theory 18 (2011), 2207-2231.
[6] B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 695-738.


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