From Statistics to Topology and Back Again

(Abstract)

We shall start by briefly discussing some statistical problems related to the structure

of the primordial universe, as seen through the Nobel Prize winning cosmic

microwave background (COBE) data.

The next step will be to turn this into an abstract problem related to the (integral and

differential) geometry generated by Gaussian random processes on manifolds.

Out of this will come extensions to Riemannian manifolds of the famous Kinematic

Fundamental Formula of classical, Euclidean, integral geometry, as well as the related

Crofton Formula.

In the end we shall see how these results shed new light on excursion probabilities for

smooth Gaussian processes, and even how they are relevant to analysing the COBE

and other astrophysical data and models.