Raphael Ponge (University of Tokyo/University of Toronto) : Noncommutative residue, conformal invariants and lower dimensional volume in Riemannian geometry

Abstract: This talk deals with the noncommutative residue trace of Wodzkicki-Guillemin and some of its applications in Riemannian geometry. Tentatively, the talk will be divided into 3 parts. In the first part, we will give an overview on the interpretation due to Connes-Moscovici of the noncommutative residue in terms of the logarithmic singularity of the kernel of a PsiDO near the diagonal, we shall present three. In the second part we will give two interesting applications of this approach. The first one is a simple and effective proof of the fact that any smoothing operators is a sum of PsiDO commutators, which can be used for proving the well-known result of Wodzicki that the noncommutative residue spans the space of traces on PsiDO's. The second one is the construction of local conformal invariants from noncommutative residue densities, which allows us to extend and simplify earlier results of Gilkey and Paycha-Rosenberg. In the third part, extending an idea of Connes, we will explain how the noncommutative residue can be used to define in a differential-geometric fashion the ``lower dimensional'' volumes of a Riemannian manifold, e.g., it can be given sense to the length and the area of Riemannian manifolds of any dimension.