Observational Constraints on the Topology (Global Geometry) of the Universe

The Universe is a physical object. Physical objects have shapes and sizes. General relativity is insufficient to describe the global shape and size of the Universe: the Hilbert-Einstein equations only treat limiting quantities towards an arbitrary point. Empirical work on measuring the shape and size of the Universe (formally: the ``3-manifold of the spatial hypersurface at constant cosmological time'', and, e.g. the ``injectivity diameter'' respectively) has progressed significantly in the late 1980's and the 1990's, using observational catalogues of galaxy clusters, of quasars and of the microwave background, though the analyses are still hindered by simplifying (and often observationally unsupported) assumptions. A review of the different observational strategies and claimed constraints was presented at the meeting.