Algebraic and Arithmetic Geometry via Stacks

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最重要的问题。在接下来的5年里，我打算实现3个主要目标：极大地推进了我与Ellenberg和Zureick-Brown一起发起的项目，以获得堆栈上有理点的渐近增长率，统一了Manin和Malle猜想。用半简单李群证明了商的一个新的平滑性准则，从而在一个长期悬而未决的波波夫问题上取得了重大进展；应用我的解决方案，我计划推广由于Vistoli的堆栈理论的一个基石结果。证明了一种新的栈变量的动机变换公式，为研究奇异变量的动机积分引入了强有力的新技术。测量代数变量上有理点的渐近增长率是算术几何的核心问题。通过最近的合作，我和Ellenberg、Zureick-Brown通过建立一个关于堆叠高度的理论，统一了该领域的两个主要猜想。此外，我们启动了一个程序，以提供对堆栈上有理点的渐近性的深入了解。我的目标是进一步证明我们的程序，证明我们的主要猜想，对于广泛的类星形堆栈，以及证明最近Lehmann， Sengupta和Tanimoto的深度结果的堆栈理论类比。群商在代数几何中是普遍存在的，特别是因为它们经常作为模空间的局部模型。在1986年ICM的演讲中，波波夫提出是否可以给出李群商光滑的标准。我的目标是概括Edidin和Whitehead最近的工作，以回答波波夫关于半简单李群的问题。作为一个应用程序，我将推广Vistoli的规范堆栈构造，以包括更大的奇点类。自引入以来，维斯托里的构造对该领域产生了深远的影响；我预计我的广义规范堆栈将为研究群商奇点提供一个新的技术行业。动力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换公式。在与Usatine合作的基础上，我的目标是将现有的变量变化公式大大推广到包括光滑的Artin堆栈。通过将这些堆栈视为奇异变量的解析，这将为计算变量上的动机积分提供一个新的技术产业。我的研究计划将在研究看似不同对象的数学家群体之间建立联系，并将在众多开放问题上取得重大进展。我的项目将培养各级HQP成为世界级的算术和代数几何研究人员，准备在自然科学和工程领域拥有非常有影响力的职业生涯。

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.