Model theory, functional transcendence, and diophantine geometry

这个项目将进一步推动最近从数学逻辑到功能超越和丢番图几何主题的令人兴奋的应用。泛函超越是研究某些函数不能被非平凡代数方程联系起来的问题。例如，函数log(x)和exp(x)之间没有非平凡的代数关系。这是由Ax得到的一个非常特殊的结果它用函数上非常简单的线性关系来描述函数和它们的指数之间的所有代数关系。利用数学逻辑的思想，我们将证明Ax结果的深远推广，涉及到各种其他映射来代替指数。丢番图几何是通过求解诸如复数等更大域的解的几何来研究方程（比如整数）的解的学科。利用数理逻辑的思想，结合上面讨论的泛函超越结果，我们将证明丢番图几何中的新结果。通常，这些结果将断言某个解只有有限多个解。在某些情况下，我们已经可以为所研究的方程证明这一点，但我们知道，即使在原则上，也没有办法找到所有的解。我们项目的一个重要方面是进一步利用数学逻辑的思想，使我们能够给出在某些情况下找到所有解的算法。

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.