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 80â€™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, I will explain how to formulate the Einstein Hilbert action in a functional analytic framework so that it can be extended to the noncommutative setting in a natural way. In the conformal case, the action density (the scalar curvature) was computed in my recent work. The new ingridents shown in the curvature formula consist of some analytic functions (related to the generating function of Bernoulli numbers). Moreover, the functions are not independent. If time permits, I will give some geometric intepretation of the functional relations.

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