相关论文: Bounded-Lipschitz Distances on the State Space of …
Topological complexity for spaces was introduced by M. Farber as a minimal number of continuity domains for motion planning algorithms. It turns out that this notion can be extended to the case of not necessarily commutative C*-algebras.…
This work provides a first step towards the construction of a noncommutative geometry for the quantum Hall effect in the continuum. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the…
Galatius and Randal-Williams defined a topology on the set of closed submanifolds of ${\mathbb R}^n$. B\"okstedt and Madsen proved that a $C^1$-version of this topology is metrizable by showing that it is regular and second countable. Using…
In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller space of a surface with boundary…
Calder\'on-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov…
Considering the Teichm\"uller space of a surface equipped with Thurston's Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest-point…
Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes's example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a…
We first prove that for all compact metrizable spaces, there exists a topological embedding of the compact metrizable space into each of the sets of compact metric spaces which are connected, path-connected, geodesic, or CAT(0), in the…
C*-algebras are widely used in mathematical physics to represent the observables of physical systems, and are sometimes taken as the starting point for rigorous formulations of quantum mechanics and classical statistical mechanics.…
We construct an isometric embedding of a bounded set in a Euclidean space into the Gromov-Hausdorff space. In particular, we can embed a bounded and connected Riemannian manifold into the Gromov-Hausdorff space by a bilipschitz map.
This contribution is an introduction to the metric aspect of noncommutative geometry, with emphasize on the Moyal plane. Starting by questioning "how to define a standard meter in a space whose coordinates no longer commute?", we list…
In this paper, we introduce the $\mathcal{F}$-metric space concept, which generalizes the metric space notion. We define a natural topology $\tau_{\mathcal{F}}$ in such spaces and we study their topological properties. Moreover, we…
The Gromov--Hausdorff distance measures the difference in shape between compact metric spaces. While even approximating the distance up to any practical factor poses an NP-hard problem, its relaxations have proven useful for the problems in…
We show that Rieffel's quantum Gromov-Hausdorff distance between two compact quantum metric spaces is not equivalent to the ordinary Gromov-Hausdorff distance applied to the associated state spaces.
We investigate local and metric geometry of weighted Carnot-Carath\'eodory spaces which are a wide generalization of sub-Riemannian manifolds and arise in nonlinear control theory, subelliptic equations etc. For such spaces the intrinsic…
We introduce fibrewise compactifications in both the setting of locally compact Hausdorff spaces and continuous maps, and the parallel setting of $C^*$-algebras and nondegenerate multiplier-valued $*$-homomorphisms. In both situations, we…
We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed…
Motivated by coarse geometry and the classical role of Roe algebras as large-scale invariants of proper metric spaces, we show that proper quantum metric spaces as introduced by Latr\'emoli\`ere are noncommutative coarse spaces. This…
A continuous bundle of $C^*$-algebras provides a rigorous framework to study the thermodynamic limit of quantum theories. If the bundle admits the additional structure of a strict deformation quantization (in the sense of Rieffel) one is…
In an earlier paper of mine relating vector bundles and Gromov-Hausdorff distance for ordinary compact metric spaces, it was crucial that the Lipschitz seminorms from the metrics satisfy a strong Leibniz property. In the present paper, for…