Spectral triples from stationary Bratteli diagrams

We construct spectral triples for path spaces of stationary Bratteli diagrams and study their associated mathematical objects, in particular their zeta function, their heat kernel expansion and their Dirichlet forms. One of the main difficulties to properly define a Dirichlet form concerns its domain. We address this question in particular in the context of Pisot substitution tiling spaces for which we find two types of Dirichlet forms: one of transversal type, and one of longitudinal type. Here the eigenfunctions under the translation action can serve as a good core for a non-trivial Dirichlet form. We find that the infinitesimal generators can be interpreted as elliptic differential operators on the maximal equicontinuous factor of the tiling dynamical system.Comment: Version 2, 49 page

Tiling groupoids and Bratteli diagrams II: structure of the orbit equivalence relation

In this second paper, we study the case of substitution tilings of R^d. The substitution on tiles induces substitutions on the faces of the tiles of all dimensions j=0, ..., d-1. We reconstruct the tiling's equivalence relation in a purely combinatorial way using the AF-relations given by the lower dimensional substitutions. We define a Bratteli multi-diagram B which is made of the Bratteli diagrams B^j, j=0, ..., d, of all those substitutions. The set of infinite paths in B^d is identified with the canonical transversal Xi of the tiling. Any such path has a "border", which is a set of tails in B^j for some j less than or equal to d, and this corresponds to a natural notion of border for its associated tiling. We define an etale equivalence relation R_B on B by saying that two infinite paths are equivalent if they have borders which are tail equivalent in B^j for some j less than or equal to d. We show that R_B is homeomorphic to the tiling's equivalence relation R_Xi.Comment: 34 pages, 14 figure

On the noncommutative geometry of tilings

This is a chapter in an incoming book on aperiodic order. We review results about the topology, the dynamics, and the combinatorics of aperiodically ordered tilings obtained with the tools of noncommutative geometry

SPECTRAL TRIPLES AND APERIODIC ORDER

International audienceWe construct spectral triples for compact metric spaces (X, d). This provides us with a new metric ¯ ds on X. We study its relation with the original metric d. When X is a subshift space, or a discrete tiling space, and d satisfies certain bounds we advocate that the property of ¯ ds and d to be Lipschitz equivalent is a characterization of high order. For episturmian subshifts, we prove that ¯ ds and d are Lipschitz equivalent if and only if the subshift is repulsive (or power free). For Sturmian subshifts this is equivalent to linear recurrence. For repetitive tilings we show that if their patches have equi-distributed frequencies then the two metrics are Lipschitz equivalent. Moreover, we study the zeta-function of the spectral triple and relate its abscissa of convergence to the complexity exponent of the subshift or the tiling. Finally, we derive Laplace operators from the spectral triples and compare our construction with that of Pearson and Bellissard

Tiling groupoids and Bratteli diagrams

Let T be an aperiodic and repetitive tiling of R^d with finite local complexity. Let O be its tiling space with canonical transversal X. The tiling equivalence relation R_X is the set of pairs of tilings in X which are translates of each others, with a certain (etale) topology. In this paper R_X is reconstructed as a generalized "tail equivalence" on a Bratteli diagram, with its standard AF-relation as a subequivalence relation. Using a generalization of the Anderson-Putnam complex, O is identified with the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram B is built from this sequence, and its set of infinite paths dB is homeomorphic to X. The diagram B is endowed with a horizontal structure: additional edges that encode the adjacencies of patches in T. This allows to define an etale equivalence relation R_B on dB which is homeomorphic to R_X, and contains the AF-relation of "tail equivalence".Comment: 34 pages, 4 figure

Cohomology and K-theory of aperiodic tilings

We study the K-theory and cohomology of spaces of aperiodic and repetitive tilings with finite local complexity. Given such a tiling, we build a spectral sequence converging to its K-theory and define a new cohomology (PV cohomology) that appears naturally in the second page of this spectral sequence. This spectral sequence can be seen as a generalization of the Leray-Serre spectral sequence and the PV cohomology generalizes the cohomology of the base space of a Serre fibration with local coefficients in the K-theory of its fiber. We prove that the PV cohomology of such a tiling is isomorphic to the Cech cohomology of its hull. We give examples of explicit calculations of PV cohomology for a class of 1-dimensional tilings (obtained by cut-and-projection of a 2-dimensional lattice). We also study the groupoid of the transversal of the hull of such tilings and show that they can be recovered: 1) from inverse limit of simpler groupoids (which are quotients of free categories generated by finite graphs), and 2) from an inverse semi group that arises from PV cohomology. The underslying Delone set of punctures of such tilings modelizes the atomics positions in an aperiodic solid at zero temperature. We also present a study of (classical and harmonic) vibrational waves of low energy on such solids (acoustic phonons). We establish that the energy functional (the "matrix of spring constants" which describes the vibrations of the atoms around their equilibrium positions) behaves like a Laplacian at low energy.Ph.D.Committee Chair: Prof. Jean Bellissard; Committee Member: Prof. Claude Schochet; Committee Member: Prof. Michael Loss; Committee Member: Prof. Stavros Garoufalidis; Committee Member: Prof. Thang L

SPECTRAL TRIPLES AND APERIODIC ORDER

International audienceWe construct spectral triples for compact metric spaces (X, d). This provides us with a new metric ¯ ds on X. We study its relation with the original metric d. When X is a subshift space, or a discrete tiling space, and d satisfies certain bounds we advocate that the property of ¯ ds and d to be Lipschitz equivalent is a characterization of high order. For episturmian subshifts, we prove that ¯ ds and d are Lipschitz equivalent if and only if the subshift is repulsive (or power free). For Sturmian subshifts this is equivalent to linear recurrence. For repetitive tilings we show that if their patches have equi-distributed frequencies then the two metrics are Lipschitz equivalent. Moreover, we study the zeta-function of the spectral triple and relate its abscissa of convergence to the complexity exponent of the subshift or the tiling. Finally, we derive Laplace operators from the spectral triples and compare our construction with that of Pearson and Bellissard

Examining Socioeconomic Inequalities in Business Schools: Empirical evidence from students’ educational and career choices

The East Coast Doctoral Conference ("ECDC") is a conference for doctoral students across management and related disciplines. Hosted annually, it is run by students in the Management PhD programs at Columbia University and NYU Stern