Rational Points on Algebraic Varieties

我的研究集中在代数变种上有理点的分布。特别是，我已经证明了，并打算证明关于代数曲面上有理点的丰度和分布的结果。到目前为止，我的大部分研究都集中在希尔伯特性质上，这是一个几何概念，表示代数变量上有理点的丰度，与zariski密度、弱弱近似和反伽罗瓦问题有关。我的主要工具是颤振理论。在第一学期，我的工作是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里发表了，但我获得了大量关于有理点和振动的一般理论的知识，这些知识后来被我应用到我的研究中。

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.