Model Theory of some Differential Equations arising from Diophantine Geometry
Lead Research Organisation:
University of Oxford
Department Name: Mathematical Institute
Abstract
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.
Organisations
People |
ORCID iD |
Jonathan Kirby (Principal Investigator) |
Publications
Bays M
(2010)
A Schanuel property for exponentially transcendental powers
in Bulletin of the London Mathematical Society
Kirby J
(2009)
The theory of the exponential differential equations of semiabelian varieties
in Selecta Mathematica
Kirby J
(2010)
Exponential algebraicity in exponential fields
in Bulletin of the London Mathematical Society
Description | The theory of the differential equations of semiabelian varieties was completely determined, extending earlier work for the split case. The usual exponential function was shown to admit a notion of dimension, precisely analogous to vector space dimension or field transcendence degree. The best possible transcendence behaviour was shown to hold for almost all power functions. |
Exploitation Route | This area of research is now a hot topic, and many people are working in it, using my results and also the knowledge and expertise which they have gained partly from me and my work. |
Sectors | Other |
Description | My research papers have been read and used by other academics working in this field. |