|TITLE:||The Transverse Geometry of Tiling Spaces|
|SPEAKER:||Jean Bellissard (Georgia Institute of Technology)|
|DATE:||Vendredi, 13 Avril 2012|
Tilings or Delone sets in R^n can be described through the concept of Hull and of Tiling Space. The former is the set of all tilings sharing with a given tiling the same family of local patches, modulo translation or modulo Euclidean isometries. The Hull is a foliated space while the Tiling Space is its canonical transversal. For tilings that are aperiodic, repetitive with finite local complexity, the Tiling Space is a Cantor set that can be constructed from a rooted tree associated with the local patch structure. All ultrametrics on the Tiling Space are built from a Michon's weight. Example will be given. Such a weighted rooted tree will give rise to a family of Spectral Triples, from which the Geometry of the Tiling Space can be reconstructed. This spectral triple gives complementary informations, like the Hausdorff dimension of the Tiling Space, or its Hausdorff measure. In addition it leads to the definition of the analog of a Laplace-Beltrami Operator, called the Pearson Laplacian. The main known properties of this operator will be given. Extension to non finite local complexity will be discussed in the conclusion.
About the speaker: Prof. Bellisard graduated from the Université Claude Bernard in 1968. He received his Ph.D. from Universite de Provence, Marseille, France 1974 where he worked until 1991. In 1991 he moved to the Université Paul Sabatier, Toulouse, where he was a chairman of the Department of Physics from 1992 till 1996. Since 1995, he has been a Senior member of the Institut Universitaire de France. In 2002 he moved to Georgia Institute of Technology in 2002. Prof. Bellisard was <ul class="MMathSStat">