Rational Points on Algebraic Varieties

我的研究重点是代数簇上有理点的分布。到目前为止，我的大部分研究都集中在Hilbert性质上，这是一个与Zariski密度、弱逼近和Galois逆问题有关的代数簇上有理点丰度的几何概念。我的主要工具是纤维理论。在我的第一个学期里，我致力于证明一类双椭圆曲面，即具有一定系数条件的对角四次曲面的Hilbert性质是满足的。不幸的是，我希望证明的结果在这段时间发表了，但我在有理点和纤维的一般理论中获得了很多知识，后来我把这些知识应用到了我的研究中。

My research centres on the distribution of rational points on algebraic varieties. In particular, I have proved and am aiming to prove results concerning the abundance and distribution of rational points on algebraic surfaces.Most of my research so far has focused on the Hilbert property, a geometric notion of the abundance of rational points on algebraic varieties with links to Zariski-density, weak weak approximation and the inverse Galois problem. My primary tool has been the theory of fibrations.During my first semester, I worked on proving the Hilbert property is satisfied for a certain class of double elliptic surfaces, namely diagonal quartic surfaces with certain coefficient conditions. Unfortunately, results which I hoped to prove were published during this time, but I gained a good deal of knowledge in the general theory of rational points and fibrations which I later applied in my research.