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.

Polilaktida oinarri duten 3D zelula-euskarri (scaffold) polimeriko biodegradagarrien diseinu, fabrikazio eta propietateak

[EU]Hiru dimentsioko inprimaketa etorkizun handiko teknologia bezala azaltzen zaigu gaur egun. Esate baterako, biomedikuntza arloan aukera berritzaileak ekar ditzake, baina baita hezkuntza, heziketa eta ikerketa munduetan ere. Teknologia berri honen abantailarik nagusiena prototipatze azkarrean datza, eta honi esker, mikro- eta makro- egitura definituak dituzten objektuak diseinatu eta fabrikatu daitezke modu lehiakorrean. Lan honen helburua 3D inprimagailu baten bitartez inprimaturiko polimero biobateragarri eta biodegradagarrietan oinarrituriko ereduen garapen eta fabrikazioan datza. Hala ere, lehenik eta behin, lehengaiak bai fisikoki eta bai termikoki karakterizatu behar dira, ondoren, 3D inprimagailuaren parametroen arteko erlazioa ezarri, eta azkenik, produktu finalaren egitura propietateak eta kalitatea aztertu. Aipaturiko lana aurrera eramateko erabili den materiala polilaktida (PLA) izan da, zeinen erabilera oso zabaldua dagoen medikuntza arloan inplante (torloju, iltze, plaka eta abar) moduan eta ehun ingeniaritzaren munduan.