  Combining analytical and numerical methods, we investigate properties of the
two-body random ensemble (TBRE). We compare the TBRE with the Gaussian
orthogonal ensemble of random matrices. Using the geometric properties of the
nuclear shell model, we discuss the information content of nuclear spectra, and
gain insight in the difficulties encountered when fitting the effective
interaction. We exhibit the existence of correlations between spectral widths
pertaining to different quantum numbers. Using these results, we deduce the
preponderance of spin-zero ground states in the TBRE. We demonstrate the
existence of correlations between spectra with different quantum numbers and/or
in different nuclei.
