Geometry of the Trace Formula

本提案的目标是继续研究轨迹公式的几何，这始于PI对基本引理的研究。特别强调的是他在之前与弗兰克尔和朗兰兹的联合工作中所做的迹公式的极限形式。PI还打算研究数论和拓扑中的相关问题，包括：Hausel关于Hitchin系统拓扑的猜想，delign - flicker关于l进局部系统数目的猜想以及与Bhargava-Shankar？S对有理椭圆曲线的平均秩做功。这些问题，除了它们固有的兴趣，应该提供洞察PI的主要项目在几何的轨迹公式。以模空间的代数几何和自同构表示为基本领域，它应该对数论和数学物理等相关数学领域产生影响。PI将培养年轻的数学家，并为他们提供解决当前数学研究中一些最令人兴奋的问题的途径。为了培养学生对数学和基础科学的兴趣，他将给学生们做关于数学的通俗演讲。

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.