Homotopical Methods in Arithmetic Geometry

数论是对结构的研究，这些结构可以广泛地称为数字系统。数字系统有非常不同的风格和广泛的应用；例如，物理世界是以实数为模型的，计算机系统是以二进制数为模型的，密码和编码方案通常是在模数的背景下设计的，等等。最近，数论和同伦理论之间形成了丰富的联系，同伦理论是受到形状和拓扑学启发的思想的代数抽象。这个项目寻求应用源自同伦理论的强大的新技术来解决关于数的突出问题。此外，一个重要组成部分将用于支持和培训年轻研究人员。更具体地说，该项目将开发一种新的“派生傅立叶分析”理论，即将傅立叶分析扩展到派生的矢量空间和动机系数。这在枚举几何，特别是算术theta函数的模性猜想，以及由Ben-Zvi-Sakellaridis-Venkatesh提出的与相对朗兰兹对偶有关的范畴周期猜想的研究中具有预期的应用。另一个方向将是研究p-进几何中的上同调运算，使用Drinfeld和Bhatt-Lurie关于棱柱上同调和上同调的新视角，着眼于解决关于Brauer群的旧问题。最后，代数K理论的方法将与Shimura Varies理论(与Galatius和Venkatesh的扩展联合工作)相结合，以更好地理解算术群和相关Galois表示的上同调。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 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.