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Widom's pseudo differential calculus and its application in heat kernel asymptotic

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Yang Liu
Ohio St. U/MPIM
Tue, 2016-10-11 14:00 - 15:00
MPIM Lecture Hall

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

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