Model theory, functional transcendence, and diophantine geometry

This project will further the recent exciting applications of ideas from mathematical logic to topics in functional transcendence and diophantine geometry.

Functional transcendence is the study of when certain functions cannot be related by non-trivial algebraic equations. For example, there is no non-trivial algebraic relation between the functions log(x) and exp(x). This is a very special instance of a result due to Ax which characterises all algebraic relations between functions and their exponentials in terms of very simple linear relations on the functions. Using ideas from mathematical logic we will prove far reaching generalizations of Ax's result involving various other maps in place of the exponential.

Diophantine geometry is the study of solutions of equations (in the integers, say) via the geometry of the solutions in larger fields such as the complex numbers. Using ideas from mathematical logic together with the functional transcendence results discussed above, we will prove new results in diophantine geometry. Typically, these results will assert that some solution has only finitely many solutions. In some instances, we can already prove this for the equations under study but we know no way, even in principle, to find all solutions. One important aspect of our project is to make further use of ideas from mathematical logic to enable us to give algorithms to find all solutions in certain cases.

Knowledge

Our project aims to develop interactions between model theory and
number theory, in particular to apply methods of o-minimality
to obtain new results in diophantine geometry. The applications of
o-minimality to diophantine problems are relatively new but have
already achieved significant results. Our project also touches on functional transcendence and differential algebraic problems,
and we expect that our project will lead to new knowledge in those
areas. Generally, our project should further encourage the use of
model-theoretic tools such as o-minimality in number theory and
cognate areas. Some of our work may lead to new knowledge
in the model theory of Pfaffian structures, concerning complexity
and effectivity issues, and may then lead to advances in applications
of these structures in some applied problems in neural networks
and economics.

People

Past interactions interactions between number theorists and model
theorists have brought new tools and problems to the attention of
both communities, to the benefit of many individual researchers.
The recent applications of o-minimality to
diophantine problems have already functioned in this way, and
we expect that our project will further contribute to
deepening and strengthening those interactions. In particular, the
project will both broaden and deepen the expertise of the researchers directly involved, especially that of the PDRAs.

Workshops

Our workshops should function as an effective and rapid forum
for the dissemination of knowledge and interaction of the
various research communities. They will both be inter-disciplinary
and will be structured to maximise discussion and
cross-fertilisation of ideas.

Society and economy

While we do not expect direct economic impact from our project,
lying as it does within pure mathematics, there may be aspects of
our work relevant to applied problems. Model theory of Pfaffian systems
is applied to problems in network complexity. Effectivity questions
in number theory are connected with fundamental distributional
questions that impact on algorithmic complexity.
We will be alive to these possibilities.