Functoriality in Function Fields

本提案的目的是在功能域的情况下研究功能化原则。本文的出发点是朗兰兹的论文《超越内窥镜》、Beilinson和Drinfeld的猜想以及最近对PI起到关键作用的基本引理的证明。这些元素之间的紧密联系将提供一条通向函数域情况下一般函数性的新途径，并将为数域情况的进一步研究提供方向。基于PI最近在证明基本引理的某些情况下的成功，这是朗兰兹计划的一个中心问题，该奖项的研究计划为代数几何和表示理论领域的年轻研究人员提供了一组新的深入和令人兴奋的问题。

The goal of this proposal is to investigate the functoriality principle in the function field case. The starting points are Langlands' paper 'Beyond Endoscopy', Beilinson and Drinfeld's conjecture and the recent proof of the Fundamental Lemma to which the PI contributed in a crucial way. Strong ties among these elements will provide a new road to general functoriality in the function field case and will give direction to further investigation in the number field case as well.Building on the PI's recent success in proving certain cases of the Fundamental Lemma, a central question in the Langlands program, the research program in this award offers a new set of deep and exciting problems for young researchers in the domain of algebraic geometry and representation theory.