Geometry of the Trace Formula

这项建议的目标是继续研究迹公式的几何，这始于Pi关于基本引理的工作。将特别强调在他与弗伦克尔和朗兰兹的前一次合作中执行的迹公式的极限形式。PI还打算研究数论和拓扑学中的相关问题，包括：Hausel关于Hitchin系统拓扑的猜想，Deligne-Flicker关于L-adic局部系统数目的猜想，以及与Bhargava-Shankar？S有关的模空间关于有理椭圆曲线的平均秩上的工作。这些问题，除了它们本身的兴趣之外，应该是Pi关于迹公式几何的主要项目。以模空间的代数几何和自同构表示为拟议活动的基本领域，它应该会对相关的数学领域，如数论和数学物理产生影响。PI将培养年轻的数学家，并让他们解决当前数学研究中一些最令人兴奋的问题。他将向学童讲授有关数学的通俗知识，以帮助他们培养对数学和基础科学的兴趣。

The goal of this proposal is to pursue the study of geometry of the trace formula, which commenced with PI's work on the fundamental lemma. A particular emphasis will be put on the limiting form of the trace formula carried out in a previous his joint work with Frenkel and Langlands. The PI also intends to work on related problems in number theory and topology including: Hausel's conjecture of the topology of Hitchin system, Deligne-Flicker's conjecture on the number of l-adic local systems and the moduli space related to Bhargava-Shankar?s work on the average rank of rational elliptic curve. These problem, in addition to their intrinsic interest, should provide insight PI's main project on the geometry of the trace formula.With algebraic geometry of moduli space and automorphic representation as basic area of the proposed activity, it should have impact on related mathematical domain as number theory and mathematical physics. The PI will train young mathematician and give them way to some of the most exciting problems in current mathematical research. He will give popular talks on mathematics to school children in order to help them acquiring taste for mathematics and fundamental sciences.