Applications of arithmetic dynamics to potential density

这个项目处理的问题是确定在数字场上定义的射影变量X何时具有所谓的势密度性质。也就是说，确定是否存在我们的数字域的有限扩展，使得X在这个更大的数字域上有许多点（或者，用更专业的语言来说，一个Zariski密集的）集合。虽然这个问题是用几何语言表述的，但它的历史悠久，比现代代数几何早了数百年。在最早的形式中，我们可以把诸如求出所有毕达哥拉斯三元组或求出佩尔方程的解的问题看作是这个问题的变体，在这些情况下，我们研究的变量是在有理数上定义的曲线，我们试图找到许多不同的整数解。

This project deals with the problem of determining when a projective variety X defined over a number field has the so-called potential density property. That is, determining whether or not there exists a finite extension of our number field such that X has many (or, in more technical language, a Zariski dense) set of points over this larger number field. Although this problem is stated in geometric language, it enjoys a long history that predates modern algebraic geometry by hundreds of years. In its earliest form, one can see problems such as finding all Pythagorean triples or the problem of finding solutions to Pell's equation as variants of this problem where the varieties one is studying are in these cases curves defined over the rational numbers and one is attempting to find many different integer solutions.