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O-minimality and diophantine geometry

O-极小性和丢番图几何

基本信息

批准号：
EP/J01933X/1
负责人：
Alex Wilkie
金额：
$ 41.74万
依托单位：
University of Manchester
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ01933X%2F1
关键词：
minimality diophantine geometry

项目摘要

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.

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）。该项目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（特别是称为o-极小的模型理论的分支）与丢番图几何（即研究具有整数或有理坐标的曲线、曲面等上的点）之间的联系。o-极小公理适用于真实的欧几里德空间（“结构”）的子集的集合，并且当满足时，暗示了各种拓扑、分析和几何有限性性质，这些性质对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，甚至是分析的流形结构的集合）来说通常是失败的。此外，许多有趣的例子o-最小的结构是已知的。威尔基是第一个注意到，有丢番图的后果集发生在o-最小的结构。皮拉的工作Diophantineproblems开始与他有影响力的1989年文件与Bjueri和他继续发展的Diophantine理论的所谓真实的analyticsets在几个方面。这最终在皮拉-威尔基定理建立了一个一般性的结果，在更广泛的设置o-极小结构。这一结果的应用，在工作的Masser，皮拉和Zannier，开辟了一个新的联系之间的数理逻辑和丢番图几何具有巨大的潜力，最显着的例子是皮拉的无条件证明的一个特殊情况下的长期存在的安德烈-奥尔特猜想。这是我们的意图进一步这样的应用，以及推进纯理论的o-最小结构铭记这一点。一个步骤的方式是一个猜想的PI关于合理的点集在一个特定的o-最小结构称为真实的指数领域。为了实现这一点，还需要对真实的Pfronan场（另一种已知为O-极小的结构）的模型理论有更精确的结果。琼斯是这两个领域的国际公认的专家，并已获得（与其他几位研究人员谁被命名为游客的建议）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，在下文中详细描述。