Mathematics

Modeled on spectral geometry of Riemannian manifolds, the asymptotic expansionof the heat operator of a geometric elliptic operator provides a functionalanalytic approach to detect  local invariants, especially, the notion of intrinsic curvature, of  noncommutative manifolds. The very first example whose computation has been carried out in great detail is the modular Gaussiancurvature on noncommutative two tori introduced in Connes and Moscovici 2014JAMS paper.  The underlying analytic problem is to compute the heatcoefficients in terms of the coefficients of the elliptic operator in questionand then study the related variational problems. The new ingredient is a familyof rearrangement operators that compress the ansatz due to the noncommutativityand between the coefficients and their derivatives. We shall show in general(beyond conformal case), that those operators can be realized by  certainhypergeometric geometric integrals and  discuss  some  combinatorial featuresderived from the hypergeometric nature.