The Trace Formula Method and the Arithmetic and Geometry of Modular Varieties in the Langlands Program

数论是数学最古老的分支之一。它研究整数的性质。数论研究对现代科学技术的影响是重大的。最值得注意的是，数论已被证明在密码学、互联网安全、电信等领域不可或缺。这是一个研究所谓的朗兰兹纲领的项目，该纲领预测了数论与其他看似不相关的数学分支之间的深层关系。朗兰兹计划的进展不仅将增进我们在数论方面的知识，而且还将证明数学的统一，表明数学中看似不同的领域受到某些共同原理的支配。这将有利于加强不同数学分支的研究人员之间的交流与合作，并将在密码学、互联网安全、电信等方面具有潜在的应用，如上所述。更详细地说，朗兰兹计划的一个中心主题是动机和自同构形式之间的互反性。最重要的工具之一是迹公式。在自守形式的背景下，存在Arthur-Selberg类型的迹公式和所谓的相对迹公式。在代数几何的背景下，存在 Grothendieck-Lefschetz-Verdier 类型的迹公式。研究人员将这些迹公式与朗兰兹纲领中自然出现的各种几何对象结合起来进行研究。这些几何对象包括Shimura簇、shtukas模空间、Rapoport-Zink空间、仿射Deligne-Lusztig簇等，它们本身就是某些几何结构的模空间。该项目的目标是在各种具体设置中描述迹公式与这些模空间的算术和几何之间的相互作用。这种理解将导致关于动机和自同构形式之间关系的结果。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Number theory is one of the oldest branches of mathematics. It studies properties of the integers. The impact of research in number theory on modern science and technology is significant. Most notably, number theory has proved indispensable in cryptography, internet security, telecommunication, and so on. This is a project to study the so-called Langlands program, which predicts deep relations between number theory and other seemingly unrelated branches of mathematics. Progress in the Langlands program will not only advance our knowledge in number theory, but also demonstrate the unification of mathematics, showing that seemingly different areas in mathematics are governed by certain common principles. This will be beneficial for enhancing communication and collaboration between researchers working in different branches of mathematics and will have potential applications to cryptography, internet security, telecommunication, and so on, as mentioned above.In more detail, a central topic in the Langlands program is the reciprocity between motives and automorphic forms. One of the most important tools is the trace formula. In the context of automorphic forms, there are trace formulas of Arthur-Selberg type and the so-called relative trace formulas. In the context of algebraic geometry, there are trace formulas of Grothendieck-Lefschetz-Verdier type. The investigator studies these trace formulas in conjunction with various geometric objects that naturally arise in the Langlands program. These geometric objects, which include Shimura varieties, moduli spaces of shtukas, Rapoport-Zink spaces, affine Deligne-Lusztig varieties, etc., are themselves moduli spaces of certain geometric structures. The goal of this project is to describe, in various concrete settings, the interaction between trace formulas and the arithmetic and geometry of these moduli spaces. Such an understanding will lead to results on the relation between motives and automorphic forms.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.