O-minimality and diophantine geometry

这个项目的调查者是亚历克斯·威尔基(曼彻斯特，派)、乔纳森·皮拉(牛津，第一协办)和加雷斯·琼斯(曼彻斯特，第一协办)。这是曼彻斯特大学和牛津大学的联合提案，曼彻斯特大学是牵头机构。该项目的目的是进一步加强数理逻辑，特别是被称为o-极小的模型理论的分支与丢番图几何(即用整数或有理坐标研究曲线、曲面等上的点)之间的联系。O-极小性公理适用于实欧几里得空间的子集(“结构”)的集合，当满足时，意味着各种拓扑、解析和几何有限性质，这些性质一般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的集合)。此外，o-极小结构的许多有趣的例子也是已知的。Wilkie是第一个注意到出现在o-极小结构中的集合存在丢番图结果的人。皮拉对丢番图问题的研究始于1989年他与Bombieri发表的有影响力的论文，然后他继续在几个维度上发展了所谓的实分析集的丢番图理论。这最终导致了Pila-Wilkie定理在更广泛的o-极小结构设置中建立了一个一般结果。这一结果在Masser、Pila和Zannier的工作中的应用，在数理逻辑和丢番图几何之间开辟了一种新的联系，具有很大的潜力，迄今为止最引人注目的例子是Pila无条件地证明了长期存在的Andre-Oort猜想的一个特例。我们打算进一步推广这种应用，并在此基础上推进o-极小结构的纯理论。其中的一步是PI关于集合上的有理点的猜想，集合上的有理点在一个特定的o-极小结构中被称为实指数域。为了实现这一点，人们还需要在真实法菲场(另一种已知为o-极小的结构)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(与提案中提到的其他几名研究人员共同)获得了迄今为止最好的结果。我们项目的所有三个方面，包括访问和研究会议的细节，在下文中都有详细描述。

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.