Asymptotic algebraic geometry and Diophantine applications

代数几何是一个丰富的数学分支，它统一了代数、拓扑学、数论和微分几何。它允许人们将看似关于代数或数字的问题重新表述为关于形状的几何问题，反之亦然。它的起源在于认识到，寻找代数方程的解或理解解的性质与这些方程所描述的代数种类的内在几何性质密切相关。朗-沃伊塔猜想是这一哲学最壮观的实现之一。一般而言，它们将方程有理解的整体行为与相应代数簇上曲率的整体行为联系起来。

Algebraic geometry is a rich branch of mathematics which unifies algebra, topology, number theory, and differential geometry. It allows one to reformulate questions which seem to be about algebra or numbers into geometrical questions about shape, or vice versa. Its origins lie in the realization that finding solutions to algebraic equations or understanding the properties of the solutions are intimately tied to the intrinsic geometric nature of the algebraic varieties those equations describe. The Lang-Vojta conjectures are one of the most spectacular realizations of this philosophy. In general terms they connect the global behaviour of rational solutions of equations to the global behaviour of the curvature on the corresponding algebraic variety.