This project concerns statistical models coming from the fields of mathematics known as number theory, combinatorics, and random matrix theory. A unifying theme in these models is that objects of central interest can be described as random fields which have a very special correlation structure in their randomness. Fields with this correlation structure are called log-correlated fields. The goal of this project is to understand the geometry of these random fields. For example, to understand how large they can be. This type of results will give us new knowledge about models of fundamental importance in number theory, combinatorics, and random matrix theory. Also these log-correlated fields are expected to be universal objects in that they appear in many models, and the methods proposed here should be general enough to apply in many cases.