Geometric Analytic Number Theory

PI将继续研究数论中渐近猜想与模空间稳定上同调之间的关系。在函数场上考虑解析数论而产生的模空间（Hurwitz空间，变异上全纯有理曲线的模空间）已经引起了拓扑学家的极大关注；事实证明，关于这些空间的有理上同调的拓扑定理和猜想，通过Lefschetz迹公式，转化成了非常清晰的渐近公式，比如“二次虚域类群中p-扭转的平均大小”。其中一些公式与数论中已建立的猜想的函数-场类似物相一致，从而证明了这些类似物；其他人则提出了新的猜想。在这个主题之外，PI将研究Kakeya问题的代数-几何方法，算术几何中展开图的应用，以及基本群幂零商的算术。PI正在两个表面上截然不同的领域之间的界面上进行研究。第一个领域是解析数论的经典主题，其核心问题涉及“计数”。例如：一个随机数不是大于1的完全平方数的倍数的概率是多少？第二个领域，更新得多，是拓扑学，它提出关于抽象形状的问题，比如弯曲的高维曲面。例如，我们可以问平面上所有n个不同点的集合的空间。事实证明，由于格罗滕迪克在20世纪60年代提出的基本见解，关于高维空间的拓扑问题可以让我们深入了解关于整数的计数问题！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这两个学科之间建立新的桥梁。PI还将继续作为数学解释者开展外联工作，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。

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.