Rational Points and Asymptotics of Distribution

当代数论的基本问题，也是过去2500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的研究探索了关于方程的解的问题（特别是，某类方程有多少解）与某类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被带到数学和数据科学中的其他问题上，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；他目前正在写一本关于几何学的书，这将涉及到一些资助的研究。该项目涵盖了数论、代数几何和数据科学的广泛问题。核心部分是PI和合作者对代数堆栈上有理点高度理论的研究。在上一个授权期间确定了该定义并研究了其性质；在当前阶段，我们将陈述一种一般启发式方法，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。这个新猜想将包括Malle猜想（有多少个数字域最多等于X?）和Batyrev-Manin猜想（给定的整数方程有多少个解，所有解最多等于X?）作为特例，但它也适用于许多新情况，甚至对经典问题也提供了新的启示。特别是，我们的工作增加了发展中的共识，即我们应该超越线束的高度，研究任意秩向量束的高度变化，开辟全新的研究方向，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（继Kaluba， Kielak和Nowak对Out（F_n）的突破之后，尝试使用“fi群”的方法来证明新群族的性质T），算术统计（证明函数场情况下关于Selmer群变分的Bhargava-Kane-Lenstra-Poonen-Rains猜想的新结果），多线性代数（理解低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它们的输入中学习对称性的行为和行为）。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.