Noncommutative Geometry Seminar

Connes-Landi noncommutative manifolds are obtained by deforming Riemannian manifolds along a torus action which have mixing Riemannian and quantum features. My current project aims to explore good notions for the intrinsic curvature for such noncommutative spaces. The pioneer work, the modular scalar curvature on noncommutative two tori was carried out independently by Connes-Moscovici and Fathizadeh-Khalkhali around 2014. My recent work (2017) generalize the computation to all even dimensional Connes-Landi manifolds. The new ingredients in the modular scalar curvature consists of actions of a noncommutative differential from the (noncommutative) metric. Such actions are defined by some interesting functions that appear in many other areas such as topology and number theory. Moreover, the functions are not independent. In this talk, I would like to report some geometric interpretation of the intriguing functional relations between them.