Asymptotics for Rational Points

本基金的研究范围在于数论与几何之间的活跃界面。几何也许是数学中最古老的部分，而数论——研究方程及其整数解的学科——也并不年轻。然而，直到最近的数学史上，研究人员才了解到这些学科是如何相互关联的，以及它们之间有多大的相互贡献。一个例子是“cap set问题”，这与流行的卡牌游戏set有关。在游戏中，有人会问：在没有合法玩法的情况下，桌上可能有多少张牌？这个问题与四维空间中点和线的几何形状有关，与数字和最后一个以3为底的数字之间的方程有关，以及它们之间的关系。2016年，PI在这个老问题上取得了重大突破，他提出的研究将继续研究导致进步的新想法，以及其他将数论和几何结合在一起的项目。该提案涵盖了数论、代数几何、拓扑学、组合学和应用数学等几个领域，与包括研究生在内的广泛研究人员合作。算术统计学的一个核心问题是：最多有多少个数字字段有区别？更具体地说：对于对称群S_n的特定子群G，它们中有多少是伽罗瓦群G ？Malle的一个著名猜想提出了该计数随X增长的渐近行为的描述。当代数论的许多主要主题（例如巴尔加瓦关于计数四次和五次扩展的工作，科恩-伦斯特拉猜想的进展）都与这个猜想有关。在之前的工作中，PI证明了函数场F_q(t)上的Cohen-Lenstra猜想可以用代数拓扑的方法逼近，使用Grothendieck的ettale上同论作为两个主题之间的桥梁。现在PI提出在K = F_q(t)的情况下证明Malle猜想的上界，再次使用拓扑和算术方法的组合，但现在使用量子洗牌代数理论的输入。在另一个项目中，PI提出研究Malle猜想与Batyrev-Manin猜想之间的类比，后者研究代数变量上有界高度的有理点的渐近性。高度是代数点复杂度的自然概念，就像数域的判别式一样。在这里，技术桥梁是代数堆栈理论；PI将发展Deligne-Mumford堆栈上有界高度的有理点理论，这首先需要定义堆栈上一个点的高度。特别地，数域的判别式是有限群的分类堆栈上的一个点的高度（在新的意义上）。PI将为堆栈制定一个广义的Batyrev-Manin猜想，该猜想专门针对Malle猜想和Batyrev-Manin猜想。PI还将研究新定义的性质：例如，PI将旨在证明法尔廷高度实际上是在这个意义上的阿贝尔变体的模堆栈上的高度。PI还提出了加性组合学、fi模块的同调性和机器学习的几何问题。

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.