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.

当代数字理论的基本问题，这也是过去两千年来数字理论的基本问题，是：我们对方程的解决方案的总体数量有何看法？ PI的研究探讨了有关方程解决方案的问题的关系（特别是某些方程式有多少解决方案），并具有有关某些空间几何形状的问题；反过来，PI拟议的研究的另一面研究了可以在数学和数据科学领域的其他问题上引起高维空间的几何形状的方式。 Pi的研究与他作为印刷，广播和社交媒体数学的普遍的工作紧密联系在一起。他目前正在研究一本关于几何学的书，该书将涉及一些资助的研究。该项目涵盖了数量理论，代数几何和数据科学的广泛问题。一个中心部分是PI和合作者在代数堆栈中理性点的高度理论的工作。该定义被固定及其在上一个授予期间研究的属性；在目前，我们将陈述一个普遍的启发式启发式方法，以实现有限身高堆栈的不对称计数点，并致力于提供新病例。这项新协议将包括特殊情况，即MALLE协议（最多有多少个数字字段）和Batyrev-Manin的猜想是有判别的（整数中最多都有多少个解决方案的解决方案？特别是，我们的工作增加了发展的共识，即我们应该超出线条捆绑包的高度，并研究任意等级的向量捆绑包的高度变化，打开了全新的研究方向，并揭示了文献的现有领域之间的新联系。 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),多线性代数（了解低坡度量张量的基因座的代数和凸几何形状）和数据科学（对流行的机器学习协议有什么作用，并且不从输入中了解对称性的研究）。该奖项反映了NSF的法定任务，并通过评估了基金会的评估，并通过评估了基金会的范围和广阔的范围。

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