Algebraic and Arithmetic Geometry via Stacks

通过堆栈学习代数和算术几何

• 批准号: RGPIN-2022-02980
• 负责人: Satriano, Matthew
• 金额: $ 2.7万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759274
• 关键词: Algebraic; Arithmetic; Geometry; via; Stacks

A long-term goal of my research career is to develop novel stack-theoretic techniques and apply these techniques to questions of fundamental importance in algebraic and arithmetic geometry. Over the next 5 years, I aim to accomplish 3 major objectives: 1. Greatly advance the program I initiated with Ellenberg and Zureick-Brown to obtain asymptotic growth rates for rational points on stacks, unifying the Manin and Malle Conjectures. 2. Prove a new smoothness criterion for quotients by semi-simple Lie groups, thereby making major headway on a longstanding open question of Popov; applying my solution, I plan to generalize a cornerstone result in stack theory due to Vistoli. 3. Prove a novel motivic change of variables formula for stacks, introducing powerful new techniques to study motivic integrals on singular varieties. Measuring the asymptotic growth rate of rational points on algebraic varieties is central to arithmetic geometry. Through recent joint work, Ellenberg, Zureick-Brown, and I have unified two major conjectures in the field by developing a theory of heights on stacks. Moreover, we initiated a program to provide great insight into asymptotics for rational points on stacks. I aim to further our program by proving our main conjecture for the wide class of horospherical stacks, as well as proving stack-theoretic analogues of recent deep results of Lehmann, Sengupta, and Tanimoto. Group quotients are ubiquitous in algebraic geometry, particularly because they often serve as local models for moduli spaces. In his 1986 ICM address, Popov asked whether one could give a criterion for when Lie group quotients are smooth. I aim to generalize recent work with Edidin and Whitehead to answer Popov's question for semi-simple Lie groups. As an application, I will generalize Vistoli's canonical stack construction to include a much larger class of singularities. Ever since its introduction, Vistoli's construction has had a profound impact on the field; I anticipate that my generalized canonical stacks will provide a new industry of techniques for studying group quotient singularities. Motivic integration has had a revolutionary impact on algebraic geometry and mathematical physics alike. Central to the theory is a change of variables formula. Based on joint work with Usatine, I aim to greatly generalize the existing change of variables formulas to include smooth Artin stacks. By viewing such stacks as resolutions of singular varieties, this will give a new industry of techniques for computing motivic integrals on varieties. My research program will build connections between communities of mathematicians studying seemingly different objects, and will make significant progress on numerous open problems. My program will train HQP at all levels to become world-class researchers in arithmetic and algebraic geometry, poised to have highly impactful careers in natural sciences and engineering.

我研究生涯的一个长期目标是开发新的堆栈理论技术，并将这些技术应用于代数和算术几何中具有根本重要性的问题。在接下来的五年里，我的目标是实现三个主要目标：1。极大地推进我与Ellenberg和Zureick-Brown一起发起的程序，以获得堆栈上合理点的渐近增长率，统一Manin和Malle猜想。2.通过半单李群证明一个新的光滑性准则，从而在波波夫的一个长期悬而未决的问题上取得重大进展;应用我的解决方案，我计划推广Vistoli在堆栈理论中的一个基石结果。3.证明了一个新的栈的动机变量变化公式，引入强大的新技术来研究奇异簇上的动机积分。测量代数簇上有理点的渐近增长率是算术几何的核心。通过最近的联合工作，Ellenberg，Zureick-Brown和我通过发展堆栈高度理论统一了该领域的两个主要理论。此外，我们启动了一个程序，以提供更好的洞察渐近合理的堆栈上的点。我的目标是进一步证明我们的主要猜想的广泛类horospheric堆栈，以及证明堆栈理论的类似物最近的深刻结果莱曼，森古普塔，和谷本。群同分在代数几何中无处不在，特别是因为它们经常作为模空间的局部模型。在他1986年的ICM演讲中，波波夫问是否可以给出一个标准，当李群同分是光滑的。我的目标是推广最近的工作与Edidin和怀特黑德回答波波夫的问题半单李群。作为一个应用，我将推广Vistoli的规范堆栈结构，以包括更大的一类奇点。自提出以来，Vistoli的构造对该领域产生了深远的影响;我预计我的广义典范堆栈将为研究群商奇异性提供一个新的技术行业。Motivic积分对代数几何和数学物理都产生了革命性的影响。该理论的核心是变量变化公式。基于与Usatine的共同工作，我的目标是大大推广现有的变量变化公式，包括光滑的Artin堆栈。通过将这些堆栈视为奇异品种的分辨率，这将为计算品种上的motivic积分提供一个新的技术产业。我的研究计划将建立数学家社区之间的联系，研究看似不同的对象，并将在许多开放的问题上取得重大进展。我的计划将培养各级HQP成为算术和代数几何的世界级研究人员，准备在自然科学和工程领域拥有极具影响力的职业生涯。