 
Summary: Part III
The Geometry of Random Fields
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You could not possibly have got this far in our book without having read the preface,
so you already know that you have finally reached the main part of this book. It is here
that the important results lie and it is here that, after over 250 pages of preparation,
there will also be new results.
WiththegeneraltheoryofGaussianprocessesbehindusinPartI, andthegeometry
of Part II now established, we return to the stochastic setting.
There are three main (classes of) results in this part. The first is an explicit formula
for the expected Euler characteristic of the excursion sets of smooth Gaussian random
fields. In the same way that we divided the treatment of the geometry into two parts,
Chapter 11 will cover the theory of random fields defined over simple Euclidean
domains and Chapter 12 will cover fields defined over Whitney stratified manifolds.
Unlike the case in Chapter 6, however, even if you are primarily interested in the
manifold scenario you will need to read the Euclidean case first, since some of the
manifold computations will be lifted from this case via atlasbased arguments.
As an aside, in the final section (Section 12.6) of Chapter 12 we shall return to a
purelydeterministicsettinganduseourGaussianfieldresultstoprovideaprobabilistic
proof of the classical ChernGaussBonnet theorem of differential geometry using
