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没有质因数的可能性有多大）在几何中有类似的问题——在这种情况下——如果不允许红点与蓝点碰撞，一组红点和一组蓝点如何在球体表面上移动？在这种情况下，一个相关的几何事实是，红点（以及蓝点）可以按照你喜欢的任何顺序重新排列，而不会违反无碰撞规则；例如，如果红蓝点位于一条线上而不是一个球体上，这就不成立了。在非技术环境中，很难说为什么这与两个数字具有相同质因数的可能性有关；我只想说，这种背景之间的转换不断地产生了数论和几何的新思想，也是PI提出的研究的中心主题。除此之外，PI在几何和机器学习的界面上有几个项目——例如，我们是否可以使用这种技术，使机器能够下非常强大的国际象棋和围棋，在网格中找到大的点配置，这样就不会有三个形成等腰三角形？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以外的社区的推广工作密切相关，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，为公众培训早期职业科学家的写作能力。该奖项还将支持研究生的研究。本研究涉及数论、代数拓扑学和代数拓扑学交界的一系列问题。一个主要目标将是利用PI与Tran和Westerland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几乎完全无法解决的老问题。新技术表明反常的束在算术统计中的一个有趣的角色，PI将在授权期间探索。PI还提出了一系列算术几何项目，包括：“扭曲”算术统计的新方向(例如：)，以及代数曲线族中Ceresa类变化的计算和理论工作。PI还将继续与工业界的研究人员合作，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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.