Geometry of Arithmetic Statistics and Related Topics

算术统计几何及相关主题

• 批准号: 2301386
• 负责人: Jordan Ellenberg
• 金额: $ 40万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2301386&HistoricalAwards=false
• 关键词: Geometry; Arithmetic; Statistics; Related; Topics

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.

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没有质因数的可能性有多大）在几何中具有类似物 - 在这种情况下 - 一组红点和一组红点如何如果不允许红点与蓝点碰撞，蓝点会在球体表面移动吗？在这种情况下，一个相关的几何事实是红点（同样是蓝点）可以按任何顺序重新排列。你喜欢在不违反无碰撞规则的情况下；例如，如果红点和蓝点位于一条线上而不是球体上，则这不会是真的。与两个数字具有共同质因数的可能性有关；足以说明这种背景之间的转变不断产生数论和几何中的新思想，并且是 PI 提出的超越研究的中心主题。那个，那个PI 在几何学和机器学习的接口上有几个项目——例如，我们是否可以使用一种技术，使机器能够下非常强的国际象棋和围棋，以找到网格中点的大型配置，使得没有三个可以形成一个等腰三角形？这是一个玩具问题，但我们将开发的技术将告诉我们很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的推广工作密切相关。 ，其中包括一个2021 年出版的几何畅销书；在资助期间，他将继续开发为公众培训早期职业科学家的项目。该奖项还将支持研究生的研究，涵盖一系列问题。数论、代数拓扑和代数拓扑的接口之一是利用 PI 与 Tran 和 Westerland 合作开发的技术来证明 Malle 的上限（在某些情况下还包括下限）。关于函数域的猜想，这是一个在数域情况下几乎完全无法解决的老问题，新技术表明反常滑轮在算术统计中的有趣作用，PI 将在授权期间进行探索。算术几何项目，包括：“扭曲”算术统计的新方向（例如：主导体有多少个 F_q(t) 的三次扩展？）以及关于变分的计算和理论工作作为代数曲线族 Ceresa 班的一员，PI 还将继续与业界研究人员合作开发机器学习技术，以促进纯数学，特别是极值组合学的进步。该奖项反映了 NSF 的法定使命，并被视为值得通过使用基金会的智力优点和更广泛的影响审查标准进行评估来支持。

项目成果

Mathematical discoveries from program search with large language models

使用大型语言模型进行程序搜索的数学发现

• DOI: 10.1038/s41586-023-06924-6
• 发表时间: 2024-01
• 期刊: Nature
• 影响因子: 64.8
• 作者: Romera-Paredes, Bernardino;Barekatain, Mohammadamin;Novikov, Alexander;Balog, Matej;Kumar, M. Pawan;Dupont, Emilien;Ruiz, Francisco J. R.;Ellenberg, Jordan S.;Wang, Pengming;Fawzi, Omar;Kohli, Pushmeet;Fawzi, Alhussein
• 通讯作者: Fawzi, Alhussein