I will take a broad view of "shape". My first example will be the shapes of
galaxy clusters, and the second will be cortical thickness, and its relation
to multiple sclerosis lesion density, all modeled as random fields. The key
tool used in the analysis is the Euler characteristic of the excursion set.
An exact expression for its expectation is found using new results in random
field theory involving Lipschitz-Killing curvatures and Jonathan Taylor's
Gaussian Kinematic Formula. The heuristic for this comes from the Nash
Embedding Theorem and an idea in geostatistics, so I will be touching on all
the main themes of this workshop.