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，也是过去两百年来数论的基本问题，是：对于整数方程的解我们能说些什么？PI 的研究探索了问题之间的关系。关于方程的解（特别是某些类型的方程有多少个解）以及某些类型的空间的几何问题，PI 提出的研究的另一方面研究了如何将高维空间的几何引入；影响数学中的其他问题PI 的研究与他作为印刷、广播和社交媒体上的数学普及者的工作密切相关；他目前正在写一本关于几何的书，其中将涉及一些内容。该项目涵盖了数论、代数几何和数据科学中的广泛问题，其中一个核心部分是 PI 和合作者对代数栈上有理点的高度理论的定义。和它的在上一个授予期间研究的属性；在当前期间，我们将陈述有界高度堆栈上渐近计数点的一般启发式，并致力于证明新的情况，作为特殊情况，马勒猜想（有多少个）。数域的判别式最多为 X？）和 Batyrev-Manin 猜想（给定的整数方程有多少个解，所有解最多为 X？），但适用于除此之外，我们的工作还为经典问题提供了新的思路，即我们应该超越线束的高度，并研究任意阶向量束的高度变化，开。提出全新的研究方向并揭示现有文献领域之间的新联系该项目还包括其他领域的广泛问题，包括群论（尝试使用“FI群”方法来证明性质T。对于新家庭群体来说，继突破Kaluba、Kielak 和 Nowak for Out(F_n)）、算术统计（证明关于函数域情况下 Selmer 群变化的 Bhargava-Kane-Lenstra-Poonen-Rains 猜想的新结果）、多线性代数（理解代数和低切片秩张量轨迹的凸几何）和数据科学（调查流行的机器学习协议学习和不学习的内容）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.

Sparsity of Integral Points on Moduli Spaces of Varieties

簇模空间上积分点的稀疏性