Spectral Invariants: Index and Noncommutative Residue

This project aims at investigating index theory problems at two different interrelated levels. First in the abstract setup of Connes spectral triples we will study the problem of finding a criterion for the meromorphic continuation property and thereby making the local index formula of Connes and Moscovici accessible to various applications. We present here a four step program that tackles this problem based on our experience with the spectral triple corresponding to a cross-product algebra. The second part of our effort will be to bring back some of the techniques from noncommutative geometry to the study of singular spaces. Analysis on many reasonable singular spaces can be carried out by means of a desingularization in form of a Lie groupoid. A Lie groupoid has an algebra of pseudo-differential operators associated to it (though not necessarily via a spectral triple). We are interested in considering certain spectral triples on the algebra A of order zero pseudo-differential operators over a groupoid G representing a singular space and relate the corresponding abstract calculus of pseudo-differential operators with the existing groupoid based invariant calculus. This idea is inspired by the work of R. Melrose, S. Moroianu, V. Nistor and E. Schrohe and others as mentioned in the proposal and our own experience with computing the cyclic cohomology for the cross-product spectral triple.

Many fundamental questions in mathematics are concerned with understanding and classifying geo- metric structures or constructing examples of geometric structures with prescribed properties. Here by geometric structures we mean specifying additional data on a higher dimensional surface aka manifolds. This additional data allows one to transport quantities like vectors along a curve on the manifold. The additional geometric data thus enables the computation of "rate of change" of various quantities along a curve. Hence there are natural differential operators on a manifold endowed with a geometry.This project studied symmetries and differential operators on a manifold related to different geo- metric situations. The goal has been to work toward existence of a given geometric structure and to discover properties that can distinguish them apart and eventually lead to their classification. One of our results provided an obstruction to the existence of a certain kind of geometry known as Cartan geometry on 5 dimensional manifolds. These class of geometries were introduced by Elle Cartan more than a century ago and we have established that a 5 dimensional manifold must satisfy a (topological) criterion to possess such a geometry. This obstruction does arise through the index of suitably chosen elliptic differential operator.We remark that the (Fredholm) index of an operator is a measure of its lack of invertibility. The index can be defined for a class of operators that includes hypoelliptic operators. Hypoellipticity is the requirement that an operator does not destroy or smooth out singularities present in its input. In general establishing hypoellipticity for an operator is tricky and computing the index can be even harder. A big part of this project was dedicated to the analysis of operators producing a very general criterion for hypoellipticity and related estimates. Our criterion then shows that that a large class of operators obtained from various geometric structures, including the so called curved BGG operators, are hypoelliptic. The work on their index is in progress and some intermediate results specially on cyclic homology under symmetries have already been published. The hypoellipticity of these operators produces K-theoretic and torsion elements. In analogy with the elliptic case we expect that these invariants would play a crucial role in future research.