Geometric Analytic Number Theory

PI将继续研究数论中的渐近性与模空间的稳定上同调之间的关系。 由函数域上解析数论的考虑而产生的模空间（Hurwitz空间，簇上全纯有理曲线的模空间）是已经吸引了拓扑学家大量注意的空间;事实证明，关于这些空间的有理上同调的拓扑定理和定理，通过莱夫谢茨迹公式，转化为非常清晰的渐近公式，比如“二次虚域的类群中p-挠的平均大小。其中一些公式与数论中已建立的公式的函数场类似物一致，从而证明了这些类似物;其他公式则提出了新的公式。 此外，PI还将研究Kakeya问题的代数几何方法、扩展图在算术几何中的应用、基本群的幂零子的算术等。PI正在对表面上截然不同的两个领域之间的界面进行研究。 第一个领域是解析数论的经典主题，其中心问题涉及“计数”。例如：一个随机数不是任何大于1的完美平方的倍数的概率是多少？ 第二个领域，更新得多，是拓扑学，它询问关于抽象形状的问题，比如弯曲的高维表面。 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，高维空间的拓扑问题可以让我们对整数的计数问题有非常深刻的了解！ PI和他的合作者正在证明新的拓扑定理，证明了数论中的一些旧命题，并提出了新的命题;这项工作将有助于在两个学科之间建立新的桥梁。 PI还将继续作为数学评论员开展外展工作，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。

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.