Tilings lectures

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Tileability (F. Bassino, A. Hashemi)

We consider a finite domain of the plane and are interested in how many way (if any) a given set of tiles can cover this domain without overlap. The proofs involve nice combinatoric and algebraic technics (e.g.  Gröbner basis).

slides1.pdf, exercises1.pdf, solutions1.pdf, slides2pdf, exercises2.pdf, exercises3.pdf

Random tilings (C. Boutillier, B. De Tillière)

We shall introduce dimer tilings and then consider to count them, namely the Kasteleyn / Temperley and Fisher's theorem and the Gessel-Viennot lemma. We then focus on measure: explicit expression for natural probability measures on dimer configurations and a hint at infinite volume Gibbs measures. We shall also see how global properties can emerge from randomness, in particular in the arctic circle theorem.

exercises1.pdf, exercises2.pdf

Symbolic dynamics (P. Guillon)

We shall first introduce one-dimensional symbolic dynamics: this is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator. We then shall extend this to the two-dimensional case, that is, to tilings. Very recent results shall be provided.

exercises1.pdf, exercises2.pdf

Substitutions (N. Bédaride)

Applying iteratively substitutions generate infinite words (or dynamical systems).. We link properties of these substitutions (primitivity, Pisot...) with the properties of the corresponding generated words (complexity, minimality, ergodicity...). We then extend this to two-dimensional substitutions and tiling spaces. In particular, we shall present a key resulst in this domain :substitutive tiling spaces are sofic.

exercises1.pdf, exercises2.pdf, solutions.pdf

Cellular automata (G. Theyssier)

We review various model of calculus (finite automata, transducers, Turing machines, tilings, cellular automata) and their power, in particular in connection with decidability questions. We shall then focus on cellular automata, first with a deterministic viewpoint - then with a probabilistic one.

slides.pdf, exercises1.pdf, exercises2.pdf

Cut and projection (Th. Fernique)

We explain how tilings can be constructed by digitizing planes in a higher dimensional space (cut and project scheme). We review some classic examples (Sturmian words, Penrose or Ammann-Beenker tilings...) as well as some general properties (repetitivity, local isomorphism class, complexity...). We then shall focus on the problem of local rules: when such a tiling is characterized only by its (finite) patterns? We review various results, including the most recent ones.

slides1.pdf, slides2.pdf, exercises.pdf

Self-assembly (F. Becker)

Self-assembly consider the way a tiling can be effectively formed by adding one tile at a time. We first present the most common models and focus on synchronization and efficiency issues. We then consider the problem of universality via simulation techniques.

slides1.pdf, exercises1.pdf, slides2.pdf, exercises2.pdf

Flip dynamics (O. Bodini, E. Rémila)

The flip is a local elementary transformation on tilings by rhombi. It endows spaces of tilings with a rich structure that is worth being better understood. Flips can also be used to define tochastic processes that can (sometimes) correct errors in tilings or sample them.

slides1.pdf, exercises1.pdf, exercises2.pdf

SageMath (T. Monteil)

We shall provide a general introduction to the (python based)  free mathematics software SageMath, a survival guide (how to get help, what is python and object-oriented programming and a review of the basic tools (symbolic functions, graphics, polynomials, matrices, number theory, etc.). We shall then illustrate the ability of SageMath on problems coming from other lectures.

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