Thesis (Ph. D. in Mathematics)--Vanderbilt University, Aug. 2008. / Title from title screen. Includes bibliographical references.

Thesis (Ph. D.)--University of Oregon, 2006. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 47-49). Also available for download via the World Wide Web; free to University of Oregon users.

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The goal of this thesis is to examine the actions of finite symmetry groups on aperiodic tilings. To an aperiodic tiling with finite local complexity arising from a primitive substitution rule one can associate a metric space, transformation groupoids, and C*-algebras. Finite symmetry groups of the tiling act on each of these objects and we investigate appropriate constructions on each, namely the orbit space, semidirect product groupoids, and crossed product C*-algebras respectively. Of particular interest are the crossed product C*-algebras; we derive important structure results about them and compute their K-theory.

Having as starting point a formula described in the paper of Baum & Douglas, [BmDg] the odd-degree component of the Chern character is is analyzed. Our presentation uses the obstruction theory definition Chern characteristic classes in order to emphasize the connections with the even-degree component (see Theorem 4.3.1) and leads to a natural justification of the fundamental property of the Chern character, i.e. of being a ring homomorphism. The reader is assumed to have some background in topological Δ-theory and algebraic topology. / Master of Science

Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / We investigate aspects of differential K-theory. In particular, we give a direct proof that the Freed-Lott differential analytic index is well defined, and a short proof of the differential Grothendieck-Riemann-Roch theorem in the setting of Freed-Lott differential K-theory. We also construct explicit ring isomorphisms between Freed-Lott differential K-theory and Simons-Sullivan differential K-theory, define the Simons-Sullivan differential analytic index, and prove the differential Grothendieck-Riemann-Roch theorem in the setting of Simons-Sullivan differential K-theory. / 2031-01-02

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We give an explicit description of the Loday assembly map on homotopy groups when restricted to a subgroup coming from the Atiyah-Hirzebruch spectral sequence. This proves and generalises a formula about the Loday assembly map on the first homotopy group that originally appeared in work of Waldhausen. Furthermore, we show that the Loday assembly map is injective on the second homotopy groups for a large class of integral group rings. Finally, we show that our methods can be used to compute the universal assembly map on homotopy.

Etant donnée une substitution de dimension 1, notée $\sigma$, nous pouvons définir l'enveloppe $\Omega_\sigma$ formant un système dynamique $(\Omega_\sigma, \R)$ où l'action de $\R$ sur les pavages est donnée par les translations. Si la substitution est primitive alors nous pouvons construire un pavage $P$ du demi-plan de Poincaré $\mathbb H_2 $ muni de sa métrique $\frac{\mathrm d x + \mathrm d y}{y^2}$. De manière analogue nous pouvons construire des enveloppes pour les actions de $N= \{ \mathbb{H}_2 \to \mathbb{H}_2, z \mapsto z +t, t\in \R\}$ et $G = \{ \mathbb{H}_2 \to \mathbb{H}_2, z \mapsto a z +b,(a,b) \in \R_+ ^* \times \R\}$ que l'on notera respectivement $X_P ^N$ et $X_{P(c)}^G$ (où $P(c)$ est le pavage colorié ligne à ligne pour rendre l'action de $G$ libre).\par En utilisant la notion de $C^* $-algèbre de groupoïde ainsi que les résultats obtenus dans l'article de Ian Putnam et Jared Anderson et via l'isomorphisme issu de l'équivalence Morita entre $C((\Xi \times \R)/\As)$ et $C(\Xi) \rtimes \Z$, nous pouvons donner la description de la $C^*$-algèbre de l'enveloppe du pavage hyperbolique en termes de générateurs et relations. Nous terminons par la description des générateurs de la $K$-théorie de $C(X_{P(c)}^G) \rtimes G.$ pour les substitutions de Fibonacci, Thue-Morse et Tribonacci / Given a one dimensional substitution $\sigma$, one can define the continuous hull $\Omega_\sigma$ for the $\R$-action given by translations and so we obtain a dynamical system $(\Omega_\sigma,\sigma)$. If the substitution we choose is primitive, then we can construct an hyperbolic tiling on Poincaré's half-plane equiped with its standard metric $\frac{\mathrm d x +\mathrm d y}{y^2}$. By analogy of the standard case, we can define two continuous hulls, denoted $X_P ^ N $ and $X_{P(c)}^G$, where $P(c)$ is a colored tiling (in such fashion that the action of $G$ is free), and the groups are denoted respectively $N= \{ \mathbb{H}_2 \to \mathbb{H}_2, z \mapsto z +t, t\in \R\}$ and $G = \{ \mathbb{H}_2 \to \mathbb{H}_2, z \mapsto a z +b,(a,b) \in \R_+ ^* \times \R\}$.\par Using Jean Renault's construction of the reduced $C^*$-algebra of a groupoid , the results of Ian Putnam and Jared Anderson and the Morita equivalence between $C((\Xi\times \R)/\As)$ and $C(\Xi) \rtimes \Z$, we describe the $C^*$-algebra of the hyperbolic tiling using generators and relations. Finally we obtain for the Fibonacci, Thue-Morse and Tribonacci substitutions the full description of the generators of $K_* (C(X_{P(c)}^G ) \rtimes G)$