Model Theory of some Differential Equations arising from Diophantine Geometry

这个项目是关于使用模型理论(数理逻辑的一个分支)的思想来回答与丢番图几何(数的几何)有关的问题。模型理论关注的是你可以用一种特定的、形式的语言来表达关于对象的东西。我们通常使用一阶语言，这种语言足够简单，我们经常可以完全理解语言中关于对象可以说的一切，但它的表现力足以捕捉到本质要点。该项目的几何对象是椭圆曲线，以及具有相加结构的高维类似物，如阿贝尔变种。学生们被教导在一条数字线上加法，这是相同的想法，除了我们有一个甜甜圈形状(椭圆曲线)或类似的东西在更高的维度上代替了一条线。该项目的主要目标是使用模型理论来研究在这些阿贝尔变种上绘制的曲线(和其他代数变种)如何相交。不是将模型理论直接应用于阿贝尔变种，而是有中间阶段。首先，我们将模型理论应用到一些与阿贝尔变种相关的微分代数对象上，然后我们使用一些复杂的解析几何将那里得到的答案与阿贝尔变种联系起来。由于涉及到许多不同的数学分支，另一个目的是用每个分支的工作人员都能理解的术语来解释整个过程！这项研究将由乔纳森·柯比·牛津在数学研究所的数学逻辑小组中进行。它将包括与牛津大学和世界各地其他数学系的许多其他人发展和分享想法。

This project is about using ideas from model theory (a branch of mathematical logic) to answer questions relating to diophantine geometry (the geometry of numbers ).Model theory is concerned with what you can say about objects in a particular, formal language. We usually use a first order language, which is simple enough that we can often get complete understanding of everything that can be said about an object in the language, but which is expressive enough to capture the essential points. The geometric objects of the project are elliptic curves, and higher-dimensional analogues such as abelian varieties, which have an additive structure. School children are taught to add on a number line, and this is the same idea except that instead of a line we have a doughnut-shape (an elliptic curve) or something similar in higher dimensions.The key aim of the project is to use model theory to study how curves (and other algebraic varieties) drawn on these abelian varieties can intersect. Rather than applying the model theory directly to the abelian varieties there are intermediate stages. First we apply the model theory to some differential algebraic objects related to the abelian varieties, and then we use some complex analytic geometry to relate the answers obtained there back to the abelian varieties.With many different branches of mathematics involved, another aim is to explain the whole process in terms which can be understood by people working in each of the branches separately!The research will be carried out by Jonathan Kirby in Oxford, in the Mathematical Logic Group of the Mathematical Institute. It will involve the development and sharing of ideas with many other people from Oxford and from other mathematics departments around the world.