O-minimality and diophantine geometry

该项目的研究人员是Alex Wilkie（曼彻斯特，PI）， Jonathan Pila（牛津，co-I）和Gareth Jones（曼彻斯特，co-I）。这是曼彻斯特大学和牛津大学的联合提议，曼彻斯特大学是牵头机构。该项目的目的是进一步联系数学逻辑，特别是被称为o-minimality的模型理论分支，和丢芬图几何（即对曲线、曲面等点的整数或有理坐标的研究）。极小性公理适用于真实欧几里得空间（“结构”）的子集集合，并且当满足时，意味着各种拓扑，解析和几何有限性质，这些性质通常不适用于主流拓扑和分析中研究的更经典的集合分类（例如那些具有可微的，甚至是解析的流形结构）。此外，还有许多有趣的o-最小结构的例子。Wilkie是第一个注意到在0最小结构中出现的集合存在丢番图结果的人。Pila对丢番图问题的研究始于他与Bombieri在1989年发表的有影响力的论文，他接着发展了所谓的多维实分析集的丢番图理论。这在Pila-Wilkie定理中达到高潮，该定理在更广泛的0 -极小结构设置中建立了一般结果。这一结果的应用，在Masser， Pila和Zannier的工作中，开辟了数学逻辑和丢芬图几何之间具有巨大潜力的新联系，迄今为止最引人注目的例子是Pila对长期存在的Andre-Oort猜想的一个特例的无条件证明。这是我们的意图，以进一步的应用，以及推进纯理论的o-极小结构与此铭记。其中一步是关于PI的一个猜想，它涉及到一个特定的0 -极小结构（称为实指数场）中集合上的有理点。为了实现这一点，我们还需要在实际普法场（另一种已知的0 -极小结构）的模型理论上得到更精确的结果。琼斯在这两个领域都是国际公认的专家，并获得了迄今为止最好的研究成果（与其他几位在提案中被命名为访问者的研究人员一起）。我们项目的所有三个方面，包括访问和研究会议的细节，在下面的文本中详细描述。

The investigators on this project are Alex Wilkie (Manchester, PI), Jonathan Pila (Oxford, co-I) and Gareth Jones (Manchester, co-I). It is ajoint proposal between the Universities of Manchester and Oxford with Manchester being the lead institution.The aim of the project is to further the links between mathematical logic, specifically the branch of model theory known as o-minimality, anddiophantine geometry (i.e. the study of points on curves, surfaces etc with integer or rational coordinates). The o-minimality axiom applies tocollections of subsets of real euclidean spaces ("structures") and, when satisfied, implies a variety of topological, analytic and geometricalfiniteness properties that fail in general for the more classical classifications of sets that are studied in mainstream topology and analysis (such as those having a differentiable, or even analytic, manifold structure). Further, many interesting examples of o-minimal structures are known.Wilkie was the first to notice that there are diophantine consequences for sets occurring in an o-minimal structure. Pila's work on diophantineproblems started with his influential 1989 paper with Bombieri and he proceeded to develop the diophantine theory of the so called real analyticsets in several dimensions. This culminated in the Pila-Wilkie theorem establishing a general result in the broader setting of o-minimal structures. The application of this result, in work of Masser, Pila and Zannier, has opened up a new connection between mathematical logic and diophantine geometry with great potential, the most remarkable example to date being Pila's unconditional proof of a special case of the long standing Andre-Oort conjecture.It is our intention to further such application as well as to advance the pure theory of o-minimal structures with this in mind. One step on the way is a conjecture of the PI concerning rational points on sets in one particular o-minimal structure known as the real exponential field. To carry this out one will also need more precise results on the model theory of the real Pfaffian field (another structure known to be o-minimal). Jones is an internationally recognised expert in both these areas and has obtained (jointly with several other researchers who are named as visitors in the proposal) the best results to date. All three aspects of our project, including details of visits and research meetings, are described at length in the following text.