Algebraic and Arithmetic Geometry via Stacks

我研究生涯的一个长期目标是开发新的堆叠理论技术，并将这些技术应用于代数和算术几何中基本重要的问题。在接下来的5年里，我的目标是实现三个主要目标：1.大大推进我与Ellenberg和Zureick-Brown共同发起的程序，以获得堆叠上有理点的渐近增长率，统一了Manin和Malle猜想。2.用半单李群证明了商的一个新的光滑性判据，从而在波波夫的一个长期悬而未决的问题上取得了重大进展；应用我的解决方案，我计划推广Vistoli在叠加论中的一个基石结果。3.证明了一种新的堆栈的动机变量变换公式，引入了强大的新技术来研究奇异变元上的动机积分。度量代数簇上有理点的渐近增长率是算术几何的核心。通过最近的合作，埃伦伯格、祖瑞克-布朗和我通过发展一种堆栈高度理论，统一了该领域的两个主要猜想。此外，我们发起了一个计划，以提供对堆栈上有理性点的渐近性的深入了解。我的目标是通过证明我们对大类天球堆叠的主要猜想，以及证明堆叠理论与Lehmann、Sengupta和TAnimoto最近的深层结果的类似来进一步推进我们的计划。群商在代数几何中是无处不在的，特别是因为它们经常被用作模空间的局部模型。在1986年的ICM演讲中，波波夫问人们是否可以给出一个关于李群商是光滑的准则。我的目标是推广最近与Edidin和Whitehead的工作，以回答波波夫关于半单李群的问题。作为一个应用程序，我将概括Vistoli的规范堆栈结构，以包含更大的奇点类别。自引入以来，Vistoli的构造对该领域产生了深远的影响；我预计我的广义正则堆栈将为研究群商奇点提供一个新的技术行业。基元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公式的改变。基于与Usatine的联合工作，我的目标是将现有的变量变换公式大大推广到包括光滑Artin堆栈在内。通过将这样的堆栈视为奇异簇的分解，这将为计算簇上的动机积分提供一个新的技术行业。我的研究计划将在研究看似不同对象的数学家社区之间建立联系，并将在许多悬而未决的问题上取得重大进展。我的计划将培训各级HQP成为世界一流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极具影响力的职业生涯。

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.