The Langlands Program is a mathematical framework that unifies questions in many different areas of mathematics, especially number theory and linear algebra. The traditional arithmetic Langlands program has been studied for more than fifty years, and has found significant applications to solving classical Diophantine equations, such as the solution of Fermat's last theorem. The geometric Langlands program, which is relatively new, is under rapid development as it is also connected with other subjects such as geometry and physics, from where one can draw intuitions. In this project, the principal investigator will explore connections between these two different facets of the Langlands philosophy by applying geometric methods to study arithmetic problems. This award will also provide support for graduate students studying topics related to this exciting but demanding area of research.
In more detail, the principal investigator will generalize his previous results on cohomological correspondences between mod p fibers of Shimura varieties, and apply them to the study of arithmetic level rising problems and the Beilinson-Bloch-Kato conjecture. Along the way, he will develop necessary ingredients such as the tame local geometric Langalands correspondence for p-adic groups. The principal investigator will also continue his study of p-adic aspects of cohomology theory with coefficients, and compare and relate them with the complex aspects and l-adic aspects.
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.
