Model theory, functional transcendence, and diophantine geometry

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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.