WILDMOD: Model Theory of wild mathematical structures, new perspectives via geometries and positive logic.

WILDMOD：狂野数学结构的模型理论，通过几何和正逻辑的新视角。

• 批准号: EP/Y027833/1
• 负责人: H Macpherson
• 金额: $ 23.84万
• 依托单位: University of Leeds
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY027833%2F1
• 关键词: WILDMOD; Model; Theory; wild; mathematical

This project will investigate wild algebraic structures through two overlapping perspectives in model theory: a geometric approach and a positive logic approach. Model theory is a branch of mathematical logic that deals with definable sets (a generalisation of the notion of algebraic variety). It identifies deep notions of tameness which draw dividing lines among mathematical structures (e.g. groups, fields, graphs...). Tame structures (for instance stable structures) are the terrain of applications and connections between model theory and other areas (e.g. algebraic and diophantine geometry). This project will push the boundaries of tameness to structures previously considered wild but ripe for model-theoretic treatment. In Workpackage 1, we describe a geometric approach for wild structures which includes real-closed fields and algebraically closed valued fields expanded by generic subgroups andgeneric homomorphisms, as well as the ring of algebraic integers. In Workpackage 2 we describe how recent rekindlement in positive model theory shall be used to work with the classes of generic difference pseudo-finite/real-closed fields, filling a gap in the model theory of difference fields.D'Elbée - the Fellow - will receive training in the model theory group in Leeds and during a 3-month secondment in Munster. The transfer of knowledge will be two-way; d'Elbée will benefit from broad expertise in Leeds (his supervisor Macpherson, also Mantova, Eleftheriou and Brooke-Taylor) and in Munster (his supervisor Hils, also Tent and Jahnke), and will transfer to the Leeds and Munster teams his knowledge of wild expansions of fields and of positive logic (through study groups, seminars and discussions). The project incorporates outreach activities for all ages at MathCity Leeds, public talks for a general audience, training in PhD supervision, a website for the research project, and a final 3-day workshop on the research themes of his project.

这个项目将通过模型论中两个重叠的观点来研究野生代数结构：几何方法和正逻辑方法。模型论是数理逻辑的一个分支，它处理可定义集合（代数簇概念的推广）。它识别了在数学结构（例如组、域、图...）之间画分界线的驯服的深层概念。驯服结构（例如稳定结构）是模型理论和其他领域（例如代数和丢番图几何）之间的应用和联系的领域。这个项目将把驯服的界限推到以前被认为是野生的结构，但模型理论的处理已经成熟。在工作包1中，我们描述了一个几何方法的野生结构，其中包括实闭域和代数闭值域扩大的一般子群和一般同态，以及环的代数整数。在工作包2中，我们描述了如何在最近的重新点燃积极的模型理论应使用的类通用差分伪有限/真正的封闭领域，填补了空白的模型理论的差异领域。D 'Elbée-研究员-将接受培训的模型理论组在利兹和3个月的借调期间在明斯特。知识的转移将是双向的; d 'Elbée将受益于利兹（他的导师Macpherson，也是Mantova，Eleftheriou和Brooke-Taylor）和明斯特（他的导师Hils，也是Tent和Jahnke）的广泛专业知识，并将向利兹和明斯特团队转移他对领域和正逻辑的疯狂扩展的知识（通过学习小组，研讨会和讨论）。该项目包括在数学城利兹所有年龄段的推广活动，面向普通观众的公开讲座，博士监督培训，研究项目网站，以及关于他的项目研究主题的最后3天研讨会。