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

这个项目将通过模型理论中的两个重叠的视角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，它处理可定义的集合(代数簇概念的推广)。它识别出在数学结构(如组、域、图等)之间划出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)之间的应用和连接的领域。这个项目将把驯服的界限推向以前被认为是狂野的，但已经成熟的模型理论处理的结构。在工作包1中，我们描述了一种用于野生结构的几何方法，该方法包括实闭域和代数闭值域，这些域由泛子群和泛同态展开，以及代数整数环。在工作包2中，我们描述了如何使用最近重新启动的正模型理论来处理类属差分伪有限/实封闭场，填补了差场模型理论的一个空白。D‘Elbée研究员将在利兹的模型理论小组和在明斯特的3个月的借调期间接受培训。知识的转移将是双向的；d‘Elbée将受益于利兹(他的主管麦克弗森，还包括Mantova，Eleftheriou和Brooke-Taylor)和明斯特(他的主管希尔斯，还有Tent和Jahnke)的广泛专业知识，并将他对领域的疯狂扩展和积极逻辑的知识转移到利兹和明斯特团队(通过学习小组、研讨会和讨论)。该项目包括在数学城市利兹为所有年龄段的人举办的外联活动、面向普通听众的公开演讲、博士指导培训、研究项目的网站，以及为期3天的关于其项目研究主题的最后研讨会。

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.