Anabelian Methods in Arithmetic and Algebraic Geometry

代数几何和数论都与多项式方程组有关。前者试图理解它们定义的形状，后者试图理解系统的整数或有理解的集合。21世纪世纪数学最重要的主题之一是这两种观点之间的相互作用：所讨论的形状的性质影响方程的算术，反过来又受到方程算术的影响。本课题通过研究代数簇的基本群的算术来理解这种基本的相互作用。具体而言，本课题的内容是（1）以几何Langlands程序、p-adic分析工具、p-adic Hodge理论为输入，分析算术基本群的表示理论，以及（2）这些结果在算术几何和代数几何中的具体应用：例如几何挠猜想，非交换Chabauty-Kim方法和截面猜想.该项目旨在扩展作者和合作者开发的新技术，以调查这些困难的开放问题。该奖项反映了NSF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。

Algebraic geometry and number theory are both concerned with systems of polynomial equations. The former seeks to understand the shapes they define, and the latter seeks to understand the set of (say) integer or rational solutions to the system. One of the most important themes of 21st century mathematics is the interplay between these two viewpoints: the properties of the shape in question influence, and are in turn influenced by, the arithmetic of the equations. This project seeks to understand this fundamental interplay through a study of the arithmetic of fundamental groups of algebraic varieties.Specifically, this project is concerned with (1) an analysis of the representation theory of arithmetic fundamental groups, taking as input the geometric Langlands program, tools from p-adic analysis, and p-adic Hodge theory, and (2) an application of these results to concrete problems in arithmetic and algebraic geometry: for example, the geometric torsion conjecture, the non-abelian Chabauty-Kim method, and the section conjecture. The project aims to expand on new techniques developed by the author and collaborators to investigate these difficult open questions.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.