Geometric and Arithmetic Langlands Program

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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 this research has resulted in significant applications to solving classical Diophantine equations, for example, the proof of Fermat's last theorem. The geometric Langlands program, which is relatively new, is under rapid development thanks to powerful tools from algebraic geometry. In this project, the investigator will explore connections between these two different facets of the Langlands program by applying geometric methods to study arithmetic topics. This award supports research training for graduate students. In more detail, this is a project to study various geometric and arithmetic aspects of the Langlands program. The investigator will attack his recent own conjectures related to the local and global Langlands correspondences. Along the way, he will develop necessary ingredients in both geometric Langlands and in number theory, such as the tame version of local geometric Langlands correspondence and the construction/study of moduli spaces of global Galois representations. The investigator will also apply the newly developed methods and tools to the study of concrete questions in representation theory and arithmetic 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.