  Recent progress in quantum field theory and quantum gravity relies on mixed
boundary conditions involving both normal and tangential derivatives of the
quantized field. In particular, the occurrence of tangential derivatives in the
boundary operator makes it possible to build a large number of new local
invariants. The integration of linear combinations of such invariants of the
orthogonal group yields the boundary contribution to the asymptotic expansion
of the integrated heat-kernel. This can be used, in turn, to study the one-loop
semiclassical approximation. The coefficients of linear combination are now
being computed for the first time. They are universal functions, in that are
functions of position on the boundary not affected by conformal rescalings of
the background metric, invariant in form and independent of the dimension of
the background Riemannian manifold. In Euclidean quantum gravity, the problem
arises of studying infinitely many universal functions.
