This book is aimed at theoretical and mathematical physicists and mathematicians interested in
modern gravitational physics. I have thus tried to use language familiar to readers working on
classical and quantum gravity paying attention both to difficult calculations and to existence
theorems and discussing in detail the current literature. The first aim of the book is to
describe recent work on the problem of boundary conditions in one-loop quantum cosmology. The
motivation of this research was to under stand whether supersymmetric theories are one-loop
finite in the presence of boundaries with application to the boundary-value problemsoccurring
in quantum cosmology. Indeed higher-loop calculations in the absence of boundaries are already
available in the litera ture showing that supergravity is not finite. I believe however that
one-loop calculations in the presence of boundaries are more fundamental in that they provide
a more direct check of the inconsistency of supersymmetric quantum cosmology from the
perturbative point of view. It therefore appears that higher-order calculations are not
strictly needed if the one-loop test already yields negative results. Even though the question
is not yet settled this research has led to many interesting new applications of areas of
theoretical and mathematical physics such as twistor theory in flat space self-adjointness
theory the generalized Riemann zeta-function and the theory of boundary counterterms in super
gravity. I have also compared in detail my work with results by other authors explaining
whenever possible the origin of different results the limits of my work and the unsolved
problems.
