Collaborative Research: Periods and Functorial Transfer

Symmetries play an important role in mathematics and in physics. This research project concerns functions that are invariant under a collection of symmetries, called automorphic forms, that are connected to number theory, representation theory, harmonic analysis and string theory. The Langlands and relative Langlands programs predict subtle relations between different spaces of automorphic forms, a structure that is closely related to many questions in number theory and analysis. In this project the PIs will work together to study the Langlands and relative Langlands programs and to extend them to new situations. The PIs will also systematically collaborate on the training of PhD students and in developing graduate-student-centered seminars for them. This project concerns functoriality and the study of periods in the Langlands and relative Langlands programs and for covering groups. The PIs, working jointly, will establish a new Shimura correspondence which is detected by a period involving a theta function. To do so, they will develop a suitable relative trace formula. Also, working jointly with Ginzburg, the PIs will study periods for the discrete, non-cuspidal spectrum, and study endoscopic lifts and periods. These projects will give new information about periods of automorphic forms and will add to the understanding of relative trace formulas. They naturally complement recent advances for reductive groups and the relative Langlands program and by including covering groups they will broaden our understanding of these topics. The PIs will also contribute to graduate training and to the nation’s development of a diverse, globally competitive STEM workforce. 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.