Geometric Langlands Program and Arithmetic Algebraic Geometry

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The PI studies the geometric Langlands program and its applications to arithmetic geometry. The PI will investigate the geometry and the cohomology of the special fibers of Shimura varieties by applying method coming from geometric representation theory, and investigate the possible relations between the characters appearing in the cohomology and the hypothetical 'character sheaves' for p-adic groups. The PI will also explore the possible relation between geometric Langlands program in positive characteristic and the mod p Langlands program.The Langlands Program is among the most important mathematical frameworks of our time. It is an attempt to unify many different areas of mathematics, such as representation theory and number theory. The traditional arithmetic Langlands program, which is directly related to numbers, has developed more than fifty years and the early development has found significant applications in solving classical diophantine equations (e.g. Fermat's last theorem). The geometric Langlands program, which is a relatively new, is under rapid development, thanks to powerful tools from algebraic geometry. The two parts of the Langlands program share a lot of similarities and techniques, while there seems no direct connection between them before. The PI will try to make some direct connections between them and try to apply some geometric methods to study some more arithmetic problems.