Geometric Analytic Number Theory

几何解析数论

• 批准号: 1101267
• 负责人: Jordan Ellenberg
• 金额: $ 29.83万
• 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: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101267&HistoricalAwards=false
• 关键词: Geometric; Analytic; Number; Theory

The PI will continue his investigations into the relationship between asymptoticconjectures in number theory and stable cohomology of moduli spaces. The moduli spaces that arise from consideration of analytic number theory over function fields (Hurwitz spaces, moduli spaces of holomorphic rational curves on varieties) are spaces that have already attracted a great deal of attention from topologists; it turns out that topological theorems and conjectures about the rational cohomology of these spaces translates, via the Lefschetz trace formula, into very clean asymptotic formulas for things like "the average size of p-torsion in the class group of a quadratic imaginary field." Some of these formulas agree with function-field analogues of established conjectures in number theory, and thus prove those analogues; others suggest new conjectures. Beyond this main theme, the PI will study algebro-geometric methods for Kakeya problems, applications of expander graphs in arithmetic geometry, and the arithmetic of nilpotent quotients of fundamental groups.The PI is carrying out research at the interface between two fields that are quite different on the surface. The first field is the classical subject of analytic number theory, whose central questions involve "counting." For example: what is the chance that a random number is not a multiple of any perfect square greater than 1? The second field, much newer, is that of topology, which asks questions about abstract shapes, like curvy high-dimensional surfaces. For instance, one can ask about the space of all sets of n different points in the plane. It turns out, thanks to fundamental insights developed by Grothendieck in the 1960s, that topological questions about high-dimensional spaces can give us very deep insights into counting questions about whole numbers! The PI and his collaborators are proving new topological theorems which prove some old conjectures in number theory, and suggest new conjectures; this work will help build new bridges between the two subjects. The PI will also continue outreach work as a mathematical expositor, maintaining a popular math blog and writing articles about mathematics for national publications.

PI将继续研究数论中的渐近猜想与模空间的稳定上同调之间的关系。 考虑函数域上的解析数论而产生的模空间（Hurwitz 空间、簇上的全纯有理曲线的模空间）已经引起了拓扑学家的广泛关注。事实证明，关于这些空间的有理上同调的拓扑定理和猜想可以通过 Lefschetz 迹公式转化为非常干净的渐近公式，例如“二次虚场类群中 p 挠场的平均大小”。 其中一些公式与数论中已建立的猜想的函数域类比一致，从而证明了这些类比；其他人提出了新的猜想。 除了这个主题之外，PI还将研究Kakeya问题的代几何方法、展开图在算术几何中的应用以及基本群幂零商的算术。PI正在两个领域之间的接口上进行研究表面上不同。 第一个领域是解析数论的经典学科，其核心问题涉及“计数”。 例如：随机数不是任何大于 1 的完全平方数的倍数的可能性有多大？ 第二个领域较新，是拓扑学领域，它提出有关抽象形状的问题，例如弯曲的高维表面。 例如，我们可以询问平面上所有 n 个不同点的集合的空间。 事实证明，得益于 Grothendieck 在 20 世纪 60 年代提出的基本见解，有关高维空间的拓扑问题可以为我们提供有关整数的计数问题的非常深入的见解！ PI和他的合作者正在证明新的拓扑定理，这些定理证明了数论中的一些旧猜想，并提出了新的猜想；这项工作将有助于在两个学科之间建立新的桥梁。 PI 还将继续作为数学讲解员开展外展工作，维护一个受欢迎的数学博客，并为国家出版物撰写有关数学的文章。