Richard B. Melrose
Respectfully dedicated to Lars Hörmander on the occasion of his sixty-fifth unbirthday
Some recent, and some new, results on the structure of algebras of pseudodifferential operators on compact manifolds with boundary are discussed. In particular their relationship to compactifications of non-compact spaces is emphasized. It is shown how these relationships allow the methods of pseudodifferential operators (``microlocalization'') to be applied to problems in scattering and spectral theory.