Arithmetic geometry of moduli spaces and applications

当想要定义多项式方程的解时，可以得到数域。它们是数论研究的主要对象。阿贝尔变分是椭圆曲线的一种推广，是代数几何的主要研究对象之一。参数化阿贝尔变量的空间是模曲线的推广，称为志村变量。志村变异在现代数论中起着核心作用。它们与伽罗瓦表示法（绝对伽罗瓦群的投影，其研究可以说是代数数论的主要目的），显式类场论（试图对数字域进行显式处理），模形式（其中的例子是编码二次型表示整数的次数的函数，并在物理和组合学中有许多应用），非阿贝耳调和分析等密切相关。

Number fields are obtained when one wishes to define solutions to polynomial equations. They are the primary object of study of number theory. Abelian varieties are a generalization of elliptic curves and are one of the primary objects of study of algebraic geometry. The spaces that parameterize abelian varieties are a generalization of modular curves, called Shimura varieties. Shimura varieties play a central role in modern number theory. They are intimately related to Galois representations (projections of the absolute Galois group, whose study, arguably, is the chief purpose of algebraic number theory), explicit class field theory (that attempts to give an explicit handle on number fields), modular forms (examples of which are theta functions which encode the number of times a quadratic form represents an integer, and have many applications to physics and combinatorics), nonabelian harmonic analysis and more.