Geometric Langlands Program and Arithmetic Algebraic Geometry

PI研究几何朗兰兹程序及其在算术几何中的应用。PI将应用几何表示理论的方法研究Shimura簇的特殊纤维的几何和上同调，并研究上同调中出现的特征标与p-adic群的假设“特征标层”之间的可能关系。PI还将探索几何Langlands程序之间的可能联系，在积极的特点和模p Langlands程序。Langlands程序是我们这个时代最重要的数学框架之一。它试图统一数学的许多不同领域，如表示论和数论。传统的与数直接相关的算术朗兰兹程序已经发展了五十多年，早期的发展在求解经典丢番图方程（如费马最后定理）中得到了重要的应用。几何朗兰兹计划，这是一个相对较新的，正在迅速发展，由于强大的工具，从代数几何。朗兰兹纲领的两个部分有许多相似之处和技术，而以前它们之间似乎没有直接的联系。PI将尝试在它们之间建立一些直接的联系，并尝试应用一些几何方法来研究一些更多的算术问题。

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.