Random subgroups of semisimple Lie groups

Northwestern University At Chicago, Evanston IL

Investigators

Abstract

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 three years and have already proven useful. Among other problems, the PI will focus on the Margulis ergodicity conjecture (a variant of a conjecture of Schoen and Yau), the Nori problem, and the Stuck--Zimmer conjecture. In particular, the PI will build on his very recent breakthroughs in the third topic. The PI's past research had an impact on various fields including geometric and combinatorial group theory, finite groups, differential geometry, algebraic number theory, automorphic forms, representation theory, topology, operator algebras, combinatorics, computer science and theoretical physics, and future impacts are expected. Specifically, progress in the Nori problem may provide tools to distinguish between thin and arithmetic groups. Such techniques are currently unavailable but required in many situations, e.g. when studying the monodromy groups of hypergeometric equations. Solving the Stuck--Zimmer conjecture will have many applications, in particular to the study of limit multiplicity theorems and Plancherel measures associated with Hilbert--Blumenthal modular varieties. The Margulis-Schoen-Yau conjecture is significant both to the study of discrete subgroups of Lie groups and to differential geometry. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.