A fundamental mathematical practice is the study and classification of all objects with specific characteristics. By itself, this is a deterministic question, hence, it is surprising that introducing some randomness into the problem can make it easier. This project will develop and use probabilistic methods to study deterministic objects with sophisticated structure, to describe their typical behavior, what types of symmetries the structures have, and what types of models can be used to detect their properties. The PI plans to use deep tools that he and collaborators have developed in recent years to address fundamental conjectures on various types of groups that describe number theoretic, geometric, or topological properties. The PI will lead activities to train future researchers in this active area, including the organization of a year-long emphasis year at Northwestern University with a spring school aimed at students and an international conference for experts, the organization of a special session at a major conference in January 2026, and the teaching of a research seminar aimed at graduate students and postdoctoral fellows. Building on the text recently completed by the PI, he will write another book on specialized topics related to this project, and building on the experience the PI has in supervising undergraduates in a research program, the PI will direct additional cohorts of students. The main techniques on which the project relies were developed in the last