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 contrast to the global aspects,

which is rooted in the homological algebra framework, the local

geometry is quite subtle. The main technical tool used in the

computation is a symbol calculus for pseudo differential operators.

The theory of pseudo differential operators plays a significant role

in global analysis. In the literature, the operators are often

presented as Fourier integral operators on local charts of manifolds,

which do not exist in the noncommutative geometry setting. It forces

us to look at a coordinate-free (intrinsic) approach. In this talk,

we will discuss such a pseudo differential calculus developed by H.

Widom around 1980 and use it to compute the scalar curvature as one of

the heat asymptotic coefficients with respect to the scalar Laplacian

operator on a closed Riemannian manifold. If time permits, we will

explain some new phenomenons when setting foot in a noncommutative

world.

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