Model theory, functional transcendence, and diophantine geometry

这个项目将进一步推动最近令人兴奋的应用程序的想法，从数学逻辑的主题，在功能超越和丢番图几何。函数超越（英语：Functional Transcendence）是研究当某些函数不能被非平凡代数方程所关联时。例如，函数log（x）和exp（x）之间没有非平凡的代数关系。这是一个非常特殊的例子，结果由于Ax的特点，所有代数关系的职能和指数方面的非常简单的线性关系的职能。利用思想从数学逻辑，我们将证明深远的推广Ax的结果涉及各种其他地图的地方指数。丢番图几何是研究解决方案的方程（在整数，说）通过几何的解决方案在更大的领域，如复数。利用数理逻辑的思想和上面讨论的函数超越性结果，我们将证明丢番图几何中的新结果。典型地，这些结果将断言某个解只有2个解。在某些情况下，我们已经可以对所研究的方程证明这一点，但我们不知道如何找到所有的解，即使在原则上也是如此。我们项目的一个重要方面是进一步利用数理逻辑的思想，使我们能够给出算法来找到某些情况下的所有解决方案。

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.