Abstract: Geometric operators defined on a compact Riemannian manifold, e.g. Laplacian, Dirac, provide a framework in which we can investigate some geometric properties while we are completely working with algebra of operators on Hilbert spaces and commutators and spectral analysis of operators. In this setting we will have objects called spectral triples introduced by Alain Connes, which will play role of differential calculus on our (noncommutative) spaces.

A spectral triple is a triple (A,H,D) in which A is an involutive algebra (plays role of $C^\infty (M)) and H is Hilbert space on which A acts continuously (it is analogous of the space of the sections of vector bundle which D acts on) and D is an operator (it is our first order elliptic differential operator) which has some properties.

This talk is the first session of a series of talks in which we will investigate different properties and examples and objects related to spectral triples. The talk will start with definition of spectral triples and we shall go through classical examples to show where the ideas come from and at the end a spectral triple defined on NC-torus will be discussed.