Noncommutative Geometry Seminar

A general question behind the talk is to explore a good notion for intrinsic curvature in the framework of noncommutative geometry started by Alain Connes in the 80's. It has only recently begun (2014)to be comprehended via the intensive study of modular geometry on the noncommutative two tori. In this talk, we will focus on a class of noncommutative manifolds obtained by deforming certain Riemannian manifolds along a torus action. I will explain how to formulate some basic notions in Riemannian geometry that are often described in local charts (such as the metric tensor, scalar curvature)using the language of functional analysis so that they will survive in the noncommutative setting. The highlight is that under a noncommutative conformal change of metric, we found not only the conformal change of the scalar curvature in Riemannian geometry but also some exciting new features: the quantum part of the curvature which is hidden in the commutative setting. What is more striking is that the quantum part of the curvature is defined by certain entire functions which play a prominent role in many other areas in mathematics (e.g. in the theory of characteristic classes).