Effective methods in o-minimality and diophantine geometry

This project will develop applications of mathematical logic to diophantine geometry.

Diophantine geometry is the study of solutions of polynomial equations in integers, or other arithmetically significant solutions, using the geometry associated with those equations. Certain types of solutions of equations can be described geometrically as "unlikely intersections". Deep conjectures predict that these unlikely intersections should be rare, for example, there should be only finitely many such solutions to a given equation.

A major goal in diophantine geometry is not only to prove that an equation has finitely many solutions, but to bound the number of solutions or to give an algorithm that finds all solutions. In this project, we will establish such effective bounds for unlikely intersections.

Techniques from mathematical logic have played an important role in the theory of unlikely intersections. We will apply recent effective bounds for the complexity of the definitions of objects from diophantine geometry. Together with recent exciting progress on effective counting theorems, these will provide the logical tools we will apply to unlikely intersections.