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Rational Points and Asymptotics of Distribution

有理点和分布渐进

基本信息

批准号：
2001200
负责人：
Jordan Ellenberg
金额：
$ 40万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001200&HistoricalAwards=false
关键词：
Rational Points Asymptotics Distribution

项目摘要

The fundamental question of contemporary number theory, which has also been the fundamental question of number theory for the last two and a half thousand years, is: what can we say about solutions to equations in whole numbers? The PI’s research explores the relation of questions about solutions to equations (in particular, how many solutions certain kinds of equations have) with questions about geometry of certain kinds of spaces; in turn, another side of the PI’s proposed research investigates the ways geometry of high-dimensional spaces can be brought to bear on other problems in mathematics and data science which might not immediately "look like" geometry. The PI’s research is closely entwined with his work as a popularizer of mathematics in print, broadcast, and social media; he is currently working on a book about geometry which will involve some of the funded research.The project covers a wide range of problems in number theory, algebraic geometry, and data science. A central part is the work of PI and collaborators on a theory of height for rational points on algebraic stacks. The definition was pinned down and its properties studied during the previous granting period; during the present period, we will state a general heuristic for asymptotically counting points on stacks of bounded height, and work towards proving new cases. This new conjecture will include as special cases the Malle conjectures (how many number fields are there of discriminant at most X?) and the Batyrev-Manin conjectures (how many solutions are there to a given equation in integers all of which are at most X?) but applies to many new cases besides, and sheds new light even on the classical questions. In particular, our work adds to the developing consensus that we should go beyond heights attached to line bundles and study the variation of heights attached to vector bundles of arbitrary rank, opening up whole new directions of research and revealing new connections between existing sectors of the literature. The project also includes a wide range of problems in other areas, including group theory (an attempt to use the method of “FI-groups” to prove property T for new families of groups, following the breakthrough of Kaluba, Kielak, and Nowak for Out(F_n)), arithmetic statistics (proving new results towards the Bhargava-Kane-Lenstra-Poonen-Rains conjectures on variation of Selmer groups in the function field case), multilinear algebra (understanding the algebraic and convex geometry of the locus of low-slice-rank tensors), and data science (investigation of what popular machine learning protocols do and don’t learn about symmetry from their input).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.

当代数论的基本问题，也是过去2500年来数论的基本问题，是：关于整数方程的解，我们能说些什么？ PI的研究探索了方程解的问题（特别是某些类型的方程有多少解）与某些类型空间的几何问题之间的关系;反过来，PI提出的研究的另一个方面是研究高维空间的几何如何能够影响数学和数据科学中的其他问题，这些问题可能不会立即“看起来像”几何。 PI的研究与他在印刷、广播和社交媒体上推广数学的工作密切相关;他目前正在撰写一本关于几何的书，其中将涉及一些受资助的研究。该项目涵盖数论、代数几何和数据科学中的广泛问题。一个核心部分是工作的PI和合作者的理论高度合理的点代数堆栈。在前一个授予期间，定义被固定下来，其性质被研究;在本期间，我们将陈述一个通用的启发式算法，用于在有界高度的堆栈上渐近计数点，并努力证明新的情况。这个新的猜想将包括作为特殊情况的Malle定理（有多少数域的判别最多X？）和Batyrev-Manin定理（一个给定的整数方程有多少个解，所有的解都不超过X？）但适用于许多新的情况下，并揭示了新的光，甚至在经典的问题。特别是，我们的工作增加了发展中的共识，我们应该超越高度重视线丛和研究的变化高度重视向量丛的任意秩，开辟了全新的研究方向，并揭示了新的连接现有部门的文献。该项目还包括其他领域的广泛问题，包括群论（继Kaluba，Kielak和Nowak对Out（F_n）的突破之后，尝试用“FI-群”的方法证明新群族的性质T），算术统计（证明了函数域情形下塞尔默群变分的Bhargava-Kane-Lenstra-Poonen-Rains定理的新结果），多线性代数（理解低切片秩张量轨迹的代数和凸几何）和数据科学（调查流行的机器学习协议从其输入中学习什么和不学习什么对称性）。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。

项目成果

期刊论文数量（4）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Intracounty modeling of COVID-19 infection with human mobility: Assessing spatial heterogeneity with business traffic, age, and race

DOI：
10.1073/pnas.2020524118
发表时间：
2021-06-15
期刊：
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
影响因子：
11.1
作者：
Hou, Xiao;Gao, Song;Patz, Jonathan A.
通讯作者：
Patz, Jonathan A.

Heights on stacks and a generalized Batyrev–Manin–Malle conjecture

堆栈高度和广义的 Batyrev–Manin–Malle 猜想