O-minimality, Diophantine geometry, and functional transcendence

这个项目将把数理逻辑的技术应用到函数超越和丢番图几何的问题上。丢番图几何是利用代数几何中的技巧，研究整数或有理数中多项式方程组的解。研究的典型问题表明，某些方程组有一组解，这些解要么是有限的，要么可以用某种特定的方式有限地描述。这种典型的结果是Faltings对Mordell猜想的证明，即亏格大于$1$的曲线只有有限多个有理点。这个项目将调查与此相关的猜测。虽然Mordell猜想已经被证明，但由此产生的猜想，如Andr\‘{e}-Oort和Zilber-Pink猜想仍然是开放的。在这些猜想的方向上已经得到了许多部分结果。该项目可能会通过考虑这些结果在不同环境中的类似情况来扩展这种部分结果。超越性理论研究自然定义的数的代数性质，特别是某些经典函数的值，如指数函数。这门学科的基本猜想是Schanuel猜想，它捕捉到了指数函数的预期超越性质。Schanuel猜想：设$z_1，ldots，z_n在$mathbb{C}上线性无关，则域$mathbb{q}(z_1，ldots，z_n，e^{z_1}，e^{z_n})$的超越度至少为$n$。这个项目可能会在不同的背景下调查Schanuel猜想的同源词，看看猜想的自然翻译在这样的背景下是否成立，以及这些同源词的后果是什么。要考虑的一种设置的一个例子是函数超越理论，其中研究了函数的代数无关性。这一领域的一个典型结果是Ax-Schanuel定理，Ax证明了Schanuel猜想的相关同源在微分场的背景下成立。项目中使用的逻辑技术来自于模型理论。模型理论和上面讨论的问题之间有许多联系。许多像Zilber-Pink这样的猜想都有模型理论的渊源。Zilber自己提出的Zilber-Pink猜想源于他对复数求幂模型理论的研究。此外，模型论中对o-极小结构的研究为解决上述问题提供了有用的途径。O-极小结构的定义特征是每个可定义集合都是区间和点的有限并。这种刻画赋予o-极小结构以显著的驯服性质，例如细胞分解定理。解决丢番图几何和函数超越中的问题所需要的就是o-极小结构的这些温顺性质。最近，Jonathan Pila，Umberto Zannier等人将o-极小应用于研究这类问题。这个项目将在这些成果的基础上再接再厉。该项目属于EPSRC逻辑与组合学和数论的研究领域。

This project will apply techniques from mathematical logic to problems of functional transcendence and Diophantine geometry. Diophantine geometry is the study, using techniques from algebraic geometry, of the solutions to systems of polynomial equations in either the integers or the rational numbers. Typical questions studied are showing that certain systems of equations have a set of solutions which is either finite or can be described finitely in some particular way. A paradigmatic result of this kind is Faltings' proof of the Mordell conjecture, that curves of genus greater than $1$ have only finitely many rational points. This project will investigate conjectures related to this. While Mordell's conjecture has been proved, conjectures arising from it, such as the Andr\'{e}--Oort and Zilber--Pink conjectures remain open. Numerous partial results in the direction of these conjectures have already been obtained. The project would look to extend such partial results, perhaps by considering analogues of these results in different settings.Transcendence theory investigates the algebraic nature of naturally defined numbers, particularly the values of certain classical functions such as the exponential function. The underlying conjecture in this subject is Schanuel's Conjecture, which captures the expected transcendence properties of the exponential function. Schanuel's Conjecture is that, given $z_1, \ldots, z_n \in \mathbb{C}$ linearly independent over $\mathbb{Q}$, the transcendence degree of the field $\mathbb{Q}(z_1, \ldots, z_n, e^{z_1}, \ldots, e^{z_n})$ is at least $n$. This project may investigate cognates of Schanuel's Conjecture in different settings, to see if the natural translations of the conjecture hold in such settings and what the consequences of such cognates are. An example of the kind of setting to be considered is functional transcendence theory, in which the algebraic independence of functions is studied. A typical result in this area is the Ax--Schanuel theorem, whereby Ax proved the relevant cognate of Schanuel's Conjecture holds in the setting of a differential field. The logical techniques to be used in the project are from model theory. There are numerous connections between model theory and the problems discussed above. Many of the conjectures like Zilber--Pink have a model-theoretic provenance. Zilber's own formulation of the Zilber--Pink conjecture arose from his investigation of the model theory of complex exponentiation. Further, the study of o-minimal structures in model theory has provided useful approaches to the kinds of problems described above. The defining characteristic of o-minimal structures is that every definable set is a finite union of intervals and points. This characterisation imbues o-minimal structures with remarkable tameness properties, for example the Cell Decomposition Theorem. It is these tameness properties of o-minimal structures which are required in addressing problems in Diophantine geometry and functional transcendence. Recent applications of o-minimality to investigating such problems have been carried out by Jonathan Pila, Umberto Zannier, and others. This project will look to build on these results.The project falls within the EPSRC research areas Logic & Combinatorics and Number Theory.