The starting point of our talk is a study of anatomical differences between controls and patients who have suffered non-missile trauma. We use a multivariate linear model at each location in space, using Hotelling's
\(T^2\) to detect differences between cases and controls. If we include further covariates in the model, Roy's maximum root is a natural generalization of Hotelling's
\(T^2\). This leads to the Roy's maximum root random field, which includes many special types of random fields: Hotelling's
\(F\), so, in effect the Roy's maximum root random field "unifies" many different random fields.
This leads to the recent advances in random fields. We describe some recent advances both in the "theory" and "application" of smooth random fields, particularly the behaviour of the maximum of a smooth random field; the accuracy of the (arguably) well-known expected Euler characteristic (EC) approximation to the distribution of the maximum of a smooth random field; an integral-geometric "recipe" for using the EC approximation; and, finally, some important recent applications of such approximations, from classical multivariate problems to perturbation models, as well as open problems. This talk is based on joint work with Keith Worsley.