Ranking in-orbit fragmentations and space objects

The future space debris environment will be dominated by the production of fragments coming from massive fragmentations. In order to identify the most relevant parameters influencing the long term evolution of the environment and to assess the criticality of selected space objects in different regions of the circumterrestrial space, a large parametric study was performed. In this framework some indicators were produced to quantify and rank the relevance of selected fragmentations on the long term evolution of the space debris population. Based on the results of the fragmentation studies, a novel analytic index, the Criticality of Spacecraft Index, aimed at ranking the environmental criticality of abandoned objects in LEO, has been devised and tested on a sample population of orbiting objects.

GROUP QUASI-REPRESENTATIONS AND INDEX THEORY

Let M be a closed connected manifold and let D be an elliptic operator on M. Let G be a discrete countable group and let [Formula: see text] be a principal G-bundle. Connes and Moscovici showed that this data defines an analytic index ind ℓ1(G)(D) ∈ K0(ℓ1(G)). If B is a unital tracial C*-algebra, we give a formula for the trace of the image of ind ℓ1(G)(D) in K0(B) under the map induced by a quasi-representation of G in B. As an application, we reprove and generalize a formula of Exel and Loring to surface groups.

A topological index theorem for manifolds with corners

We define an analytic index and prove a topological index theorem for a non-compact manifold M0 with poly-cylindrical ends. Our topological index theorem depends only on the principal symbol, and establishes the equality of the topological and analytical index in the group K0(C*(M)), where C*(M) is a canonical C*-algebra associated to the canonical compactification M of M0. Our topological index is thus, in general, not an integer, unlike the usual Fredholm index appearing in the Atiyah–Singer theorem, which is an integer. This will lead, as an application in a subsequent paper, to the determination of the K-theory groups K0(C*(M)) of the groupoid C*-algebra of the manifolds with corners M. We also prove that an elliptic operator P on M0 has an invertible perturbation P+R by a lower-order operator if and only if its analytic index vanishes.

The analytic index of elliptic pseudodifferential operators on a singular foliation

In previous papers ([1, 2]) we defined the C*-algebra and the longitudinal pseudodifferential calculus of any singular foliation (M,). In the current paper we construct the analytic index of an elliptic operator as a KK-theory element, and prove that this element can be obtained from an “adiabatic foliation” on M×ℝ, which we introduce here.