Rational Points on Algebraic Varieties

王9700781本奖项资助以下两个不同项目的研究。（项目1）1974年，S. Lang提出了一个猜想，将数域上定义的代数变量的几何（双曲性）与该变量的算术（模态性）联系起来。这个猜想对于曲线和阿贝尔变量的子变量是成立的。其他品种鲜为人知。Sarnak教授和Wang教授已经证明，一些超曲面要么违反了Brauer- Manin阻碍无法解释的Hasse原理，要么违反了上述Lang猜想。王教授将继续研究这个问题对于0个1度循环。（课题2）对代数变种上有理点的密度进行了广泛的研究。王教授计划研究弱逼近、Brauer—Manin障碍和改进的Mazur猜想在Hurwitz族上的有理点拓扑，更一般地说，在Hurwitz空间上。由于每一种变化都被Hurwitz空间统一化，本研究将在一般情况下阐明这些问题。本研究属于数论的一般数学领域。它涉及整数代数方程的解和推广。数论是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学原因而追求它。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等各种应用领域不可或缺的工具。

Wang 9700781 This award funds research into the following two distinct projects. (Project 1) In 1974, S. Lang made a conjecture which connects the geometry (hyperbolicity) of an algebraic variety defined over a number field with the arithmetic (Mordellicity) of the variety. The conjecture is true for curves and subvarieties of Abelian varieties. Little is known for other varieties. Professors Sarnak and Wang have shown that some hypersurfaces yield either a violation of the Hasse principle which is not accounted for by the Brauer--Manin obstruction or a violation of the above Lang's conjecture. Professor Wang will continue to investigate this problem for 0-cycles of degree 1. (Project 2) The densities of rational points on algebraic varieties have been studied extensively. Professor Wang plans to study weak approximation, Brauer--Manin obstruction and the modified Mazur's conjecture on the topology of rational points on the Hurwitz families, and more generally, on Hurwitz spaces. Since every variety is uniformized by Hurwitz spaces, this research will shed light on these problems in the general case. This research falls into the general mathematical field of Number Theory. It concerns the solutions to algebraic equations in whole numbers and generalizations. Number theory is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century, it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.