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Context This project will be carried out in the Inria team Mocqua, which focuses on emerging models of computation, in particular the interaction between discrete computations and continuous mathematics. Assignment Tilings are colorations of the plane submit to local constraints. We are looking for mathematical structures capturing the communication that is needed between cells to produce valid tilings respecting the given constraints. The goal will be to define such structures, study their properties and develop techniques to establish whether a given communication structure is able to generate tilings respecting a given set of constraints. Main activities The student will read the literature to get acquainted with the field of symbolic dynamics. He/she will study in details the recent articles about local generation of tilings and of languages [1,2,3,4]. There are several research directions. One of them is to improve the results developed in [1,2] by extending finite input alphabets to countably infinite ones, and establish whether the classification presented in [2] still holds under this relaxation. Most of the results in [1] apply to weakly mixing tilings: another direction is to understand what happens beyond this case, and to invent new techniques applicable with more generality. to identify the communication structures in graph colorings. In [4], it is shown that when the number of colors is sufficiently large, the graph itself suffices to generate is colorings, but we need to understand what happens for small numbers of colors. [1] Favereau, Hoyrup. Local generation of tilings. Ergodic theory and dynamical systems. 45(11), 2025 [2] Favereau, Hoyrup. Local generation of tilings: the even bicolor Wang tilesets. Preprint, 2025 [3] Hoyrup. Local generation of languages. Preprint, 2026. [4] Hoyrup. When a graph generates its own colorings. Preprint, 2026. Skills The student must have a background in mathematics or theoretical computer science, and should show interest in at least one of the following fields, which are at the heart of the project: symbolic dynamics, combinatorics, topology.
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