The Langlands Conjectures for Connected and Disconnected Groups

这个项目研究了朗兰兹程序的各个方面——一个数学领域，连接数论和几何，以系统的方式检查代数方程解中固有的对称性。通过将这些对称与几何联系起来，该程序创建了一种强大的获取算术信息的方法。该项目特别关注明确获取此类信息。对数字运算的深刻而明确的了解，反过来对过去几十年我们今天认为理所当然的许多技术进步至关重要，包括加密和数字安全。更详细地说，该项目将建立一个统一和明确的精炼局部朗兰兹对应的构造，对于一般连接的约化p进群，在一个温和的扩展上分裂，所有的超尖参数，残差特征素数到Weyl群的阶。该项目进一步旨在将这种结构的结果与其他更一般但不太明确的结构进行比较，例如Genestier-Lafforgue的局部功能域结构。还将考虑断开连接的组的作用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project investigates facets of the Langlands program -- an area of Mathematics bridging number theory and geometry that examines in a systematic way the symmetries inherent in solutions of algebraic equations. By relating these symmetries to geometry, the program creates a powerful way of obtaining arithmetic information. The project is particularly concerned with obtaining such information explicitly. Deep and explicit knowledge of the arithmetic of numbers is in turn essential for many technological advances over the last decades that we take for granted today, including encryption and digital security.In more detail, the project will establish a uniform and explicit construction of the refined local Langlands correspondence, for general connected reductive p-adic groups splitting over a tame extension and all supercuspidal parameters, in residual characteristic prime to the order of the Weyl group. The project further aims to compare the result of this construction with other more general but less explicit constructions, such as the one of Genestier-Lafforgue for local function fields. The role of disconnected groups will also be considered.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.