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.
Diego Armentano is a professor at the Instituto de Estadística (Universidad de la República, Uruguay). His research lies at the intersection of probability theory and numerical analysis, with a particular focus on random algebraic geometry and the complexity of numerical algorithms.
Jean-Marc Azaïs is an emeritus professor at the Institut de Mathématiques de Toulouse, Université de Toulouse, France. He has worked on various topics including statistics, probability and geometry of random fields. He has also authored many articles and directed many PhDs. In 2009, he authored a monograph on geometry of random fields with Mario Wschebor.
Federico Dalmao is a professor at the Departamento de Matemática y Estadística, Universidad de la República, Salto, Uruguay. The focus of his research is on the geometric properties of stochastic processes and random fields.
Yohann De Castro, a professor at Institut Camille Jordan, École Centrale Lyon, specializes in the intersection of statistics, probability, and optimization. His research focuses on stochastic and high-dimensional geometry, convex and stochastic optimization, and latent position models.
Céline Delmas is a researcher at the French National Research Institute for Agriculture, Food and Environment (INRAE) in the Applied Mathematics and Informatics Unit. She is a statistician whose research focuses on Gaussian fields, statistical inference and modeling.
José Rafael León Ramos is a mathematician specializing in probability theory and mathematical statistics. He is a professor at the Universidad de la República (Uruguay) and was a former full professor at the Universidad Central de Venezuela. His research focuses on stochastic processes, Gaussian fields, and statistical inference. He has published extensively and is the co-author of several monographs, including volumes in the Springer series, Lecture Notes in Statistics.
Ernesto Mordecki is a professor at the Centro de Matemática, Universidad de la República, Montevideo. He earned his PhD at the Steklov Mathematical Institute (Moscow, 1994) under Albert Shiryaev, specializing in optimal stopping, Lévy processes, and mathematical finance. In addition, in 2015, he was elected to the Academy of Sciences of Uruguay.
