Geometry of Arithmetic Statistics and Related Topics

当代数论的一个基本主题是，关于数字的问题(例如：两个随机选择的大数有多大可能没有质数因子)在几何上有相似之处--在这种情况下--如果没有红点与蓝点发生碰撞，一组红点和一组蓝点如何在球体表面移动？此设置中的一个相关几何事实是，红点(以及蓝点)可以按您喜欢的任何顺序重新排列，而不会违反无碰撞规则；例如，如果红点和蓝点位于一条线上，而不是位于球体上，则不会出现这种情况。在非技术背景下，很难说出为什么这与两个数字具有相同素因数的可能性有任何关系；可以说的是，上下文之间的这种段落一直在数论和几何中产生新的想法，并且是PI提议的研究的中心主题。除此之外，PI在几何学和机器学习的界面上有几个项目--例如，我们能否使用那种使机器能够下非常强的国际象棋的技术，并在网格中找到大的点配置，从而使三个不形成等腰三角形？这是一个玩具问题，但我们开发的技术将告诉我们许多关于使用机器学习技术加速纯数学进步的前景。PI的研究与他在学术数学之外的社区推广工作紧密交织在一起，其中包括2021年出版的一本关于几何的畅销书；在资助期间，他将继续开发项目，为公众培训职业生涯早期的科学家写作。该奖项还将支持研究生的研究。所提出的研究涵盖了数论、代数拓扑学和代数拓扑学的一系列问题。一个主要目标将是利用Pi与Tran和Westland合作开发的技术来证明函数域上Malle猜想的上界(在某些情况下是下界)，这是一个在数域情况下仍然几乎完全无法解决的老问题。这些新技术表明，倒置数据在算术统计中扮演了一个有趣的角色，PI将在授权期内进行探索。PI还提出了一系列算术几何的项目，包括：“扭曲”算术统计的新方向(例如：F_q(T)的多少个三次扩张与素导体？)除了在代数曲线家族中Ceresa类的变化方面的计算和理论工作外，PI还将继续与工业研究人员合作开发机器学习技术，使之能够在纯数学，特别是极值组合学方面取得进展。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

A fundamental theme of contemporary number theory is that questions about numbers (for instance: how likely is it that two randomly chosen large numbers have no prime factors) have analogues in geometry — in this case — how can a set of red dots and a set of blue dots move around the surface of a sphere if no red dot is ever allowed to collide with a blue dot? One relevant geometric fact in this setting is that the red dots (and equally so the blue dots) can be rearranged in whatever order you like without ever breaking the no-collisions rule; this would not, for instance, be true if the red and blue dots were located on a line instead of on a sphere. In a non-technical setting it isn’t easy to say why this has anything to do with the likelihood of two numbers having a prime factor in common; suffice it to say that this passage between contexts has consistently generated new ideas in both number theory and geometry, and is a central theme of the PI’s proposed research. Beyond that, the PI has several projects at the interface of geometry and machine learning -- for example, can we use the kind of techniques that enable machines to play very strong chess and Go to find large configurations of points in in a grid such that no three form an isosceles triangle? This is a toy problem but the techniques we develop will tell us a lot about the prospects for accelerating progress in pure mathematics using machine learning techniques. The PI’s research is closely entwined with his work in outreach to the community outside academic mathematics, which includes a best-selling book on geometry published in 2021; during the funding period he will continue developing programs to train early-career scientists in writing for the public. This award will also support graduate student research. The proposed research covers a range of problems at the interface of number theory, algebraic topology, and algebraic topology. One main goal will be exploiting the techniques developed in PI’s collaboration with Tran and Westerland to prove upper bounds (and in some cases lower bounds) for Malle’s conjecture over function fields, an old problem which in the number field case remains almost entirely inaccessible. The new techniques suggest an interesting role for perverse sheaves in arithmetic statistics, will the PI will explore over the granting period. The PI also proposes a range of projects in arithmetic geometry, including: new directions in “twisted” arithmetic statistics (for instance: how many cubic extensions of F_q(t) are there with prime conductor?) and computational and theoretical work on the variation of the Ceresa class in families of algebraic curves, The PI will also continue collaborative work with researchers in industry on the development of machine learning techniques adapted to enable progress in pure mathematics, especially extremal combinatorics.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.