This mathematical research project is in the general area of the Langlands program. The Langlands program is a large research program which provides a vast blueprint for research combining several areas of mathematics. A key area of study in this project is real groups which are the fundamental symmetries occurring in nature. The question this project addresses is to understand the representations of real groups, i.e., the various ways these symmetries can be concretely exhibited. In particular, a key goal is to understand the irreducible unitary representations which are the atoms of the theory. All other representations can be constructed from these atoms. We propose to make use of geometric methods, those of Hodge theory, to completely settle this decades old question. The project has a companion part which studies symmetries closely related to number theory and physics. This constitutes another aspect of the Langlands program.

2023

2027