**Special PACM Colloquium**

In this survey talk, we describe some what might be described as the geometric theory of smooth (marginally stationary) Gaussian random functions. Beginning at the local level, the celebrated Kac-Rice formulacan be used to derive accurate approximations to the distribution of the maximum of such random functions. From this local calculation,global Riemannian integral invariants appear. Zooming out to a global level,the appearance of these integral invariants can be explained by appealing to the Kinematic Fundamental Formula.

Time permitting, we describe recent applications of the Kac-Rice formula high-dimensional statistical inference: http://arxiv.org/abs/1308.3020.These applications demonstrate the promise this local tool holds for inference about critical points for random fields that are not marginally stationary.