Please note special day, time and location.  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.