This book presents the modern theory of the geometrical characteristics of random fields and explores their interdisciplinary applications. The first five chapters concentrate on the theoretical and mathematical foundations of the expected measure of level sets, including critical points and Morse theory. They provide a streamlined proof of the Kac-Rice formula, adapted to non-Gaussian cases, and address the problem of the finiteness of moments. The text balances pedagogical explanations with recent, powerful mathematical results. Chapter 4 notably offers an accessible presentation of Hermite representation, the diagram formula, and the fourth moment theorem to establish central limit theorems for the measure of the level set, intentionally avoiding overly complex tools like Malliavin calculus. The latter chapters demonstrate the practical application of these tools across domains such as high-dimensional statistics, theoretical physics, optics and the study of critical points, concluding with a comprehensive bibliographic review. This monograph will be useful for students and researchers in probability theory, geometry, and applied sciences. It will equip them with powerful geometric tools and explicit examples to solve modern problems involving random fields.
