Differential groupoids and their application to the theory of spacetime singularities (Q1604950)

The spacetime singularities are studied by using the concept of differential groupoid (DG) in the category of structured spaces. The DG is represented by an algebra of operators on a bundle of Hilbert spaces defined on the groupoid fibers. It is shown that any spacetime with singularities can be regarded as a noncommutative space. Information on a given singularity is obtained from the isotropy group of the ``singular fiber''. This group is thought as measuring the ``strength'' of the singularity. Different properties are reflected in the structure of the representation of the corresponding DG in a Hilbert space and, to some extent, in a noncommutative algebra defined on this groupoid. Some illustrative examples, showing the ``desingularization'' process, are given.