Asymptotics for Rational Points

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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.