Model Theory of some Differential Equations arising from Diophantine Geometry

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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.