Homotopical Methods in Arithmetic Geometry

数论研究的结构可以被广泛地称为数系统。数字系统有非常不同的风格，有各种各样的应用；例如，物理世界以实数为模型，计算机系统以二进制数为模型，密码和编码方案通常在模数的背景下设计，等等。近年来，数论与同伦理论之间形成了丰富的联系，同伦理论是受形状和拓扑研究启发的思想的代数抽象。本项目旨在应用源自同伦理论的强大新技术来解决关于数字的突出问题。此外，一个重要组成部分将专门用于支持和培训青年研究人员。更具体地说，该项目将发展一种新的“派生傅立叶分析”理论，将傅立叶分析扩展到派生向量空间和动力系数。这在计数几何，特别是算术函数的模块化猜想，以及与相对朗兰兹对偶有关的范畴周期猜想（如Ben-Zvi - Sakellaridis - Venkatesh所表述的）的研究中有预期的应用。另一个方向将是研究p进几何中的上同调运算，利用由Drinfeld和bhat - lurie带来的关于棱镜和对号上同调的新视角，着眼于解决关于Brauer群的老问题。最后，为了更好地理解算术群的上同调和相关的伽罗瓦表示，代数k理论的方法将与Shimura变分理论（扩展与Galatius和Venkatesh的联合工作）相结合。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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 the study of structures which may be broadly called number systems. Number systems come in very different flavors and have a wide variety of applications; for example, the physical world is modeled on real numbers, computer systems are modeled on binary numbers, cryptographic and encoding schemes are often designed in the context of modular numbers, etc. Recently, a rich connection has formed between number theory and homotopy theory, which is an algebraic abstraction of ideas inspired by the study of shapes and topology. This project seeks to apply powerful new techniques originating in homotopy theory to resolve outstanding questions about numbers. In addition, a significant component will be devoted to the support and training of young researchers. In more specific terms, the project will develop a new theory of “derived Fourier analysis”, an expansion of Fourier analysis to derived vector spaces and motivic coefficients. This has anticipated applications in enumerative geometry, especially to modularity conjectures for arithmetic theta functions, and in the investigation of categorical period conjectures pertaining to relative Langlands duality, as formulated by Ben-Zvi – Sakellaridis – Venkatesh. Another direction will be the study of cohomology operations in p-adic geometry, using new perspectives on prismatic and syntomic cohomology due to Drinfeld and Bhatt-Lurie, with an eye towards resolving old questions about Brauer groups. Finally, methods of algebraic K-theory will be combined with the theory of Shimura varieties (extending joint work with Galatius and Venkatesh) in order to better understand the cohomology of arithmetic groups and related Galois representations.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.