Other Mathematics Commons^™

Geometry In Spectral Triples: Immersions And Fermionic Fuzzy Geometries, Luuk S. Verhoeven

Electronic Thesis and Dissertation Repository

We investigate the metric nature of spectral triples in two ways.

Given an oriented Riemannian embedding i:X->Y of codimension 1 we construct a family of unbounded KK-cycles i!(epsilon), each of which represents the shriek class of i in KK-theory. These unbounded KK-cycles are further equipped with connections, allowing for the explicit computation of the products of i! with the spectral triple of Y at the unbounded level. In the limit epsilon to 0 the product of these unbounded KK-cycles with the canonical spectral triple for Y admits an asymptotic expansion. The divergent part of this expansion is known and …

On Spectral Invariants Of Dirac Operators On Noncommutative Tori And Curvature Of The Determinant Line Bundle For The Noncommutative Two Torus, Ali Fathi Baghbadorani

Electronic Thesis and Dissertation Repository

We extend the canonical trace of Kontsevich and Vishik to the algebra of non-integer order classical pseudodifferntial operators on noncommutative tori. We consider the spin spectral triple on noncommutative tori and prove the regularity of eta function at zero for the family of operators $e^{th/2}De^{th/2}$ and the couple Dirac operator $D+A$ on noncommutative $3$-torus. Next, we consider the conformal variations of $\eta_{D}(0)$ and we show that the spectral value $\eta_D(0)$ is a conformal invariant of noncommutative $3$-torus. Next, we study the conformal variation of $\zeta'_{|D|}(0)$ and show that this quantity is also a conformal invariant of odd noncommutative tori. This …