This programme will focus on the study of the Langlands functoriality conjecture, including the endoscopy theory and the cases beyond endoscopy.
For the endoscopy case, Langlands functoriality is established by using the trace formula approach in general. Some special cases can also be established by the converse theorem-integral representation approach, and the automorphic integral transform approach.
Based on those successful cases, several new approaches are proposed. The key idea is to construct certain appropriate automorphic kernel functions and study them in various ways in order to establish functorial transfers for automorphic forms in the relative general setting.
These bring us the following four topics of the program:
- Refined structures and properties for endoscopy theory: local and global.
- Various types of trace formulas, generalized Fourier transforms and Poisson summation formulas, and applications.
- Explicit constructions of certain Langlands functorial transfers via integral transformations.
- Extension of the existing theory in the Langlands program to covering groups.
The goal of the program is to bring together experts researching in automorphic kernel functions to foster interaction, collaboration and the exchange of ideas on the new approaches. It aims to develop those approaches that will provide us further insights and progress, and lead to an eventual resolution of the Langlands functoriality conjectures.