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将继续他对数量理论中渐近转换之间的关系的研究与模量空间的稳定共同体学之间的关系。 考虑到功能领域的分析数理论（Hurwitz空间，品种上的Holomorthic Ronical曲线的模量空间）而产生的模量空间是已经引起了拓扑师的广泛关注的空间；事实证明，关于这些空间的合理共同体的拓扑定理和猜想通过lefschetz痕量公式转化为非常干净的渐近公式，例如“二次假想领域的阶级p-torsion的平均大小”。 这些公式中的一些与数量理论中既定猜想的功能场类似物一致，从而证明了这些类似物。其他人建议新的猜想。 除了这个主要主题之外，PI还将研究针对Kakeya问题的代数几何方法，在算术几何形状中的扩展器图应用以及基本组的Nilpotent商的算术。表面不同。 第一个领域是分析数理论的经典主题，其中心问题涉及“计数”。 例如：随机数不是大于1的完美正方形的倍数的机会是什么？ 第二个领域是拓扑结构，它询问了有关抽象形状的问题，例如弯曲的高维表面。 例如，可以询问平面中所有n个不同点的所有集合的空间。 事实证明，由于Grothendieck在1960年代开发的基本见解，有关高维空间的拓扑问题可以使我们对计算有关整数的问题非常深入见解！ PI和他的合作者正在证明新的拓扑定理，这些定理证明了数字理论的一些旧猜想，并提出了新的猜想。这项工作将有助于在两个主题之间建造新的桥梁。 PI还将继续担任数学扩展程序，维护流行的数学博客，并撰写有关国家出版物数学的文章。