Math-Info 2010

Towards new interactions between mathematics and computer science


Invited Seminars

Thematic month

The five weeks

Seminar On Tilings And Substitutions

This seminar is organized by Shigeki Akiyama.

Slow and Friendly: I prefer informal, slow and (possibly) blackboard using seminar. Welcome everyone interested.

1.  Application of automata to topology of tilings (March 18-19)

  • Aim: Topology of self-affine tilings is described by contact graph and adding machine. We start with an introductory account and discuss on boundary of tiles, topology, and possibly cancellation problem of free group endomorphism. Related automata problems are discussed in detail.
  • Place: FRUMAM Seminar Hall (access map), close to Marseille St. Charles Station.
  • Thursday, March 18
    • 13:30-14:00 Shigeki Akiyama: Why automata in tiling ? (Introduction)
    • 14:15-15:15 Jörg Thuswaldner: Neighbors of Rauzy fractals and disk-likeness.
    • 15:30-16:00 Jacques Sakarovitch: Rational sets in groups. (Introduction)
      • For the purpose of the study of rational sets in the free group, basic definitions and properties of automata and rational sets in non free monoids in general and in groups in particular will be introduced in this short lecture.
    • 16:15- Discussion and/or talks of interest
  • Friday, March 19
    • 9:30-10:30 Jacques Sakarovitch: Rational sets in the free group.
      • There are many connections between group and automata theory. To begin with: the natural correspondence between recognisable and finite index sets, rational and finitely generated subgroups. This talk will be an introduction to this now classical subject. The main topic will be the characterisation of rational sets in the free group via Benois's Theorem: the set of reduced words of a rational set in a free monoid over a symmetrical alphabet is a rational set.
    • 11:00-12:00 Teichmuller seminar: Alexey Glutsyuk: Aucun billiard planaire n'a d'ouvert d'orbites périodiques quadrangulaires.
      • Considérons le problème de Dirichlet pour le Laplacien dans un domaine borné d'un espace Éuclideen. Le comportement asymptotique du nombre de valeurs propres plus petites qu'un nombre N donné, quand N tend vers l'infini, a été étudié par H. Weyl, qui a trouvé le terme asymptotique principal. En 1980 V. Ivrii a trouvé le second terme asymptotique, sous la condition supplémentaire, que les orbites périodiques du billard correspondant ont mesure nulle. Il a conjecturé, que cette condition géometrique est toujours vraie. En 1989 M. Rychlik a démontré que les orbites triangulaires dans un billiard planaire ont mesure nulle. Nous avons démontré le même énoncé pour les orbites quadrangulaires. Ceci est un travail en commun avec Yuri Kudryashov.
    • 14:00-15:00 Valérie Berthé: Jacobi-Perron substitutions and boundary free group automorphisms.
    • 15:20-15:50 Shigeki Akiyama: Boundary parametrization.
    • 16:00-16:30 Tarek Sellami: Common Geometry of Substitutions with the same incidence matrix.

2.  Self-affine tiling and its dynamics (April 1-2)

  • Aim: Self-affine tiling dynamical system is a natural extension of substitution sequence dynamical system. Famous open conjecture in this field is intimately related to dilation constant appears in these systems (Pisot conjecture). Introductory surveys and discussion are expected in the seminars.
  • Place: CIRM, Annex 1, Marseille, (Access).
  • Registration for meals at CIRM is compulsory (meals are offered to all participants).
  • Thursday, April 1
    • 9:30-10:30 Pierre Arnoux: Geometric models for Pisot substitution: The duality construction.
      • In this lecture, I will explain the dual extension of substitution, and its use for building a stepped surface approximating the contracting hyperplane of a Pisot Substitution. I will show how the Pisot conjecture can be reformulated on this framework.
    • 11:00-12:00 Jeong-Yup Lee: On substitution Delone sets on Rd -- pure point spectra, algebraic coincidence and Meyer property, I
    • 14:00-15:30 Pierre Arnoux: Higher dimensional extensions of substitutions: beyond the Pisot case.
      • In this lecture, I will define the higher dimensional extensions of substitutions and their dual maps, and I will explain how these higher dimensional extensions an be used to give geometric models for hyperbolic free group automorphisms. There are a small number of worked-out examples, and many open problems in this direction.
  • Friday, April 2
    • 10:00-11:30 Jeong-Yup Lee: On substitution Delone sets on Rd -- pure point spectra, algebraic coincidence and Meyer property, II
    • 14:00-15:30 Giovani Forni: Exposants de Kontsevich-Zorich: -- Quelques résultats pour mesures SL(2,R)-invariantes non canoniques
    • 16:00-16:30 Nicolas Bédaride: Dual billiard and tiling of the plane.
      • In this short talk, I will define the dual billiard map outside convex polygon. In some cases this map gives a tiling of the plane, and the symbolic dynamics is related to substitution.

3.  Spectrum of substitutive dynamical systems (April 20-23)

  • Aim: Host showed that the eigenfunction of primitive substitutive dynamical system is continuous under mild conditions. This forms a basis to study the spectrum of substitutive dynamical system. We wish to discuss this continuity and relation to spectrum in detail.
  • Place: CIRM, Marseille.
  • Registration for meals at CIRM is compulsory (meals are offered to all participants).
  • Tuesday, April 20
    • 11:00-12:00 Sébastien Labbé: Complexité palindromique des codages de rotations et conjectures (Ernest seminar)
    • afternoon: free discussion
  • Wednesday, April 21: free discussion
  • Thursday, April 22: free discussion
  • Friday, April 23
    • 11:00-17:00 N. Pytheas Fogg: Invertible substitutions

4.  Pisot conjecture and substitution (April 28-30)

  • Aim: In symbolic substitution dynamics, Pisot conjecture is one of the most famous unsettled problems, which asserts that the dynamical system generated by irreducible Pisot substitution is pure discrete. We shall study the current situation and discuss many related problem on substitutions.
  • Place: CIRM, Chapel, Marseille, (Access).
  • Registration for meals at CIRM is compulsory (meals are offered to all participants).
  • Wednesday 28 Apr
    • 9:30-10:30 Bernd Sing: The interplay of Pisot substitutions with geometry and algebraic number theory.
    • 11:00-12:00 Fabien Durand: Eigenvalues of finite rank dynamical systems I.
    • 14:00-15:00 Dirk Frettloeh: Substitutions and their duals.
    • 15:30-16:30 Wolfgang Steiner: Multiple tilings defined by symmetric beta-transformations.
  • Thursday 29 Apr
    • 9:30-10:30 Bernd Sing: Cobounds/height and reducibility: What makes Pisot substitutions so special?
    • 11:00-12:00 Fabien Durand: Eigenvalues of finite rank dynamical systems II.
    • 14:00-15:00 Pierre Liardet: Skew products: worked examples.
  • Friday 30 April
    • Free discusssion

5.  Uploaded slides

    • As this seminar is blackboard oriented, lecturers are not obliged to prepare slides. However if some of the lecturers could supply the handwritten notes (or in any formats), we welcome and keep records of our seminar here.

    For any technical question or comment regarding this wiki, you can contact the wemaster.