Problems when generating virtual models representing real objects: Hondarribia walls

[EN] This paper is based in the following project:

On reconstruction theorems in noncommutative Riemannian geometry

We present a novel account of the theory of commutative spectral triples and their two closest noncommutative generalisations, almost-commutative spectral triples and toric noncommutative manifolds, with a focus on reconstruction theorems, viz, abstract, functional-analytic characterisations of global-analytically defined classes of spectral triples. We begin by reinterpreting Connes's reconstruction theorem for commutative spectral triples as a complete noncommutative-geometric characterisation of Dirac-type operators on compact oriented Riemannian manifolds, and in the process clarify folklore concerning stability of properties of spectral triples under suitable perturbation of the Dirac operator. Next, we apply this reinterpretation of the commutative reconstruction theorem to obtain a reconstruction theorem for almost-commutative spectral triples. In particular, we propose a revised, manifestly global-analytic definition of almost-commutative spectral triple, and, as an application of this global-analytic perspective, obtain a general result relating the spectral action on the total space of a finite normal compact oriented Riemannian cover to that on the base space. Throughout, we discuss the relevant refinements of these definitions and results to the case of real commutative and almost-commutative spectral triples. Finally, we outline progess towards a reconstruction theorem for toric noncommutative manifolds.

[A_Ribera_SanEsteban] Modelo virtual de la iglesia de San Esteban de Ribera de Valderejo (Álava)

[ES] Ribera es un pueblo abandonado dentro del parque natural de Valderejo, el único edificio que queda en pie es la antigua iglesia. La razón por la que este edificio no fue derribado es la presencia de un conjunto de pinturas murales de época medieval. Por lo demás, el edificio (de unos 25 x 12 metros en planta) se encontraba en un mal estado de conservación (de hecho, un par de años más tarde de la realización de este trabajo se derrumbó la sacristía).