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。数论的研究对现代科学技术的影响是重大的。最值得注意的是，数论在密码学、互联网安全、电信等领域已被证明是不可或缺的。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无关的数学分支之间的深刻关系。朗兰兹纲领的进展不仅将推进我们在数论方面的知识，而且还将证明数学的统一性，表明数学中看似不同的领域都受到某些共同原则的支配。这将有助于加强不同数学分支研究人员之间的交流和合作，并将在密码学，互联网安全，电信等领域有潜在的应用，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守形式之间的相互作用。最重要的工具之一是跟踪公式。在自守形式的范围内，有亚瑟-塞尔伯格型迹公式和所谓的相对迹公式。在代数几何中，存在Grothendieck-Lefschetz-Verdier型迹公式。研究人员研究这些迹公式与自然出现在朗兰兹程序的各种几何对象。这些几何对象，其中包括志村品种，模空间的shtukas，Rapoport-Zink空间，仿射Deligne-Lusztig品种，等等，都是具有特定几何结构的模空间。这个项目的目标是描述，在各种具体的设置，迹公式和这些模空间的算术和几何之间的相互作用。这样的理解将导致动机和自守形式之间的关系的结果。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.