Asymptotics for Rational Points

这项授权的研究范围在于数论和几何之间的活跃界面。几何可能是数学中最古老的部分，而数论，即研究方程及其整数解的学科，并不年轻。然而，只有在最近的数学史上，研究人员才明白这些学科是如何相互关联的，以及它们必须提供给彼此多少。一个例子是与流行的纸牌游戏套装有关的“帽子套装问题”。在游戏中，有人会问：在没有合法游戏的情况下，桌上可能有多少张牌？结果表明，这个问题涉及到4维空间中点和线的几何，涉及到数与其最后一个基数-3位数之间的方程，以及它们之间的关系。2016年，PI是这个老问题重大突破的一部分，他提出的研究将继续调查导致进步的新想法，以及其他混合了数论和几何的项目。该提案涵盖了数论、代数几何、拓扑学、组合学和应用数学的几个领域，并与包括研究生在内的一大批研究人员合作。算术统计学的核心问题之一是：判别式最多有多少个数域？更具体地说：其中有多少对对称群S_n的指定子群G有伽罗瓦群G？Malle的一个著名猜想描述了随着X的增长这一计数的渐近行为。当代数论中的许多主要主题(例如，Bhargava关于计数四次和五次扩展的工作，关于Cohen-Lenstra猜想的进展)都涉及到这种猜想的情况。在前人的工作中，PI证明了函数域F_q(T)上的Cohen-Lenstra猜想可以用代数拓扑的方法来逼近，而Grothendieck上同调理论是连接这两个问题的桥梁。现在PI建议在K=F_q(T)的情况下证明Malle猜想的上界，同样是使用拓扑学和算术方法相结合的方法，但是现在有量子洗牌代数理论的输入。在另一个项目中，PI建议研究Malle猜想和Batyrev-Manin猜想之间的相似性，后者研究代数簇上高度有界的有理点的渐近性。高度是代数点复杂性的自然概念，就像数字域的判别式一样。在这里，技术桥梁是代数堆栈理论；PI将发展出Deligne-Mumford堆栈上有界高度的有理点的理论，这首先需要定义堆栈上一点的高度。特别地，数域的判别式是有限群的分类堆栈上的点的高度(在新的意义上)。PI将为堆栈建立一个广义的Batyrev-Manin猜想，它专门针对Malle猜想和Batyrev-Manin猜想。PI还将研究新定义的性质：例如，PI将旨在证明在这个意义上Faltings高度实际上是阿贝尔变种模数堆叠上的高度。PI还提出了加法组合学、FI-模的同调和机器学习的几何等问题。

The research scope of this grant lies on the active interface between number theory and geometry. Geometry is perhaps the oldest part of mathematics, and number theory, the study of equations and their solutions in whole numbers, is hardly younger. Yet it is only in the very recent history of mathematics that researchers have understood just how interrelated these subjects are and how much they have to offer each other. One example is the "cap set problem," related to the popular card game Set. In the game, one asks: how many cards is it possible to have on the table with no legal play? It turns out that this problem has to do with the geometry of points and lines in 4-dimensional space, with equations among numbers and their last base-3 digit, and the relation between these. In 2016 the PI was part of a major breakthrough on this old problem, and his proposed research will continue investigating the new ideas that led to progress as well as other projects mixing number theory and geometry. The proposal covers several areas in number theory, algebraic geometry, topology, combinatorics, and applied math, in collaboration with a wide group of fellow researchers, including graduate students. One of the central questions of arithmetic statistics is: how many number fields are there of discriminant at most X? More particularly: how many of these have Galois group G for a specified subgroup G of a symmetric group S_n? A famous conjecture of Malle proposes a description for the asymptotic behavior of this count as X grows. Many of the major themes in contemporary number theory (e.g. Bhargava's work on counting quartic and quintic extensions, progress on Cohen-Lenstra conjectures) concern cases of this conjecture. In previous work, the PI showed that the Cohen-Lenstra conjecture over the function field F_q(t) could be approached by the methods of algebraic topology, using Grothendieck's theory of etale cohomology as the bridge between the two subjects. Now the PI proposes to prove the upper bound in the Malle conjecture in the case K = F_q(t), again using a combination of topological and arithmetic methods, but now with input from the theory of quantum shuffle algebras. In another project, the PI proposes to investigate the analogy between Malle's conjectures and the Batyrev-Manin conjectures, which study the asymptotics for rational points with bounded height on algebraic varieties. The height is a natural notion of complexity of an algebraic point just as the discriminant is for a number field. Here, the technical bridge is the theory of algebraic stacks; the PI will develop a theory of rational points of bounded height on Deligne-Mumford stacks, which first of all requires defining the height of a point on a stack. In particular, the discriminant of a number field is the height (in the novel sense) of a point on the classifying stack of a finite group. The PI will formulate a generalized Batyrev-Manin conjecture for stacks, which specializes to both Malle's conjecture and the Batyrev-Manin conjecture. The PI will also investigate properties of the new definition: for instance, the PI will aim to prove that the Faltings height is actually height on the moduli stack of abelian varieties in this sense. The PI also proposes problems in additive combinatorics, the homology of FI-modules, and the geometry of machine learning.