This monograph is devoted to a completely new approach to geometric
problems arising in the study of random fields. The groundbreaking
material in Part III, for which the background is carefully prepared in
Parts I and II, is of both theoretical and practical importance, and
striking in the way in which problems arising in geometry and
probability are beautifully intertwined.
The three parts to the monograph are quite distinct. Part I presents a
user-friendly yet comprehensive background to the general theory of
Gaussian random fields, treating classical topics such as continuity and
boundedness, entropy and majorizing measures, Borell and Slepian
inequalities. Part II gives a quick review of geometry, both integral
and Riemannian, to provide the reader with the material needed for Part
III, as well as providing some new results and new proofs of known
results along the way. Topics such as Crofton formulae, curvature
measures for stratified manifolds, critical point theory, and tube
formulae are covered. In fact, this is the only concise, self-contained
treatment of all of the above topics, which are necessary for the study
of random fields. The new approach in Part III is devoted to the
geometry of excursion sets of random fields and the related Euler
characteristic approach to extremal probabilities.
