O-minimality, Diophantine geometry, and functional transcendence

这个项目将应用从数学逻辑到功能超越和丢番图几何问题的技术。丢芬图几何是利用代数几何的技术，研究整数或有理数多项式方程组的解的一门学问。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。这类的一个典型结果是Faltings对Mordell猜想的证明，即大于$1$的属曲线只有有限多个有理点。这个项目将调查与此相关的猜想。虽然莫德尔猜想已被证明，但由此产生的猜想，如安德尔·奥尔特猜想和齐尔伯·平克猜想仍然存在。在这些猜想的方向上已经得到了许多不完全的结果。该项目将寻求扩展这样的部分结果，也许通过考虑在不同的环境中类似的这些结果。超越理论研究自然定义数的代数性质，特别是某些经典函数的值，如指数函数。这门课的基本猜想是Schanuel猜想，它抓住了指数函数的期望超越性质。Schanuel的猜想是，给定$z_1, \ldots， z_n \在$\mathbb{C}$中线性独立于$\mathbb{Q}$，域$\mathbb{Q}$ (z_1, \ldots, z_n, e^{z_1}, \ldots, 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研究Schanuel猜想的同源词，看看这个猜想的自然翻译是否在这样的背景下成立，以及这些同源词的后果是什么。要考虑的一类设置的一个例子是泛函超越理论，其中研究了函数的代数独立性。这方面的一个典型结果是Ax—Schanuel定理，其中Ax证明了Schanuel猜想的相关同系词在微分场的设置下成立。项目中使用的逻辑技术来自模型理论。在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来源。Zilber自己对Zilber- Pink猜想的表述源于他对复指数模型理论的研究。此外，模型理论中o-极小结构的研究为上述各种问题提供了有用的方法。0极小结构的定义特征是每一个可定义集合是区间和点的有限并。这种特性赋予了零最小结构显著的驯服性，例如细胞分解定理。在处理丢番图几何和功能超越问题时，需要的正是o-极小结构的这些驯服性质。最近，Jonathan Pila、Umberto Zannier等人将o-minimality应用于研究此类问题。本项目将以这些成果为基础。该项目属于EPSRC的逻辑与组合学和数论研究领域。

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.