A review is presented of some recent progress in spectral geometry on
manifolds with boundary: local boundary-value problems where the boundary
operator includes the effect of tangential derivatives; application of
conformal variations and other functorial methods to the evaluation of
heat-kernel coefficients; conditions for strong ellipticity of the
boundary-value problem; fourth-order operators on manifolds with boundary;
non-local boundary conditions in Euclidean quantum gravity. Many deep
developments in physics and mathematics are therefore in sight.Comment: 31 pages, plain Tex. Paper prepared for the Fourth Workshop on
Quantum Field Theory under the Influence of External Conditions, Leipzig,
September 199
