Geometric Langlands Program and Arithmetic Algebraic Geometry

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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 problems. In more detail, this is a project to study the geometric Langlands program and its applications to arithmetic geometry. The investigator will apply his results on the geometric Satake correspondence for p-adic groups and the p-adic Riemann-Hilbert correspondence to investigate the mod p and p-adic geometry of Shimura varieties. The investigator will also explore the relations between residues that appear in the theory of automorphic forms and the topology of spaces of rational maps.