Geometric and Combinatorial Configurations in Model Theory

模型理论中的几何和组合配置

• 批准号: 431667816
• 负责人: Professor Dr. Martin Hils
• 金额: --
• 依托单位: Institut de Mathématiques de Jussieu-Paris Rive Gauche (UMR 7586)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/431667816?language=en
• 关键词: Geometric; Combinatorial; Configurations; Model; Theory

GeoMod is a Collaborative Research project between France and Germany. Contemporary model theory studies abstract properties of mathematicalstructures from the point of view of first-order logic. It tries to isolate combinatorial properties of definable sets such as the existence of certain configurations, or of rank functions, and to use them to obtain structural consequences. These may be algebraic or geometric in nature, and can be applied to specific structures such as Berkovich spaces, difference-differential algebraic geometry, additive combinatorics or Erdős geometry.A good example of a configuration implying algebraic structure is the group configuration theorem which asserts that certain combinatorial patterns are necessarily induced by a group, and that moreover the structure of the groups which may give rise to this configuration is highly restricted. This result, which generalizes the coordinatization theorems of geometric algebra was given its definitive form for stable theories by Hrushovski in 1986 and hereafter became one of the most powerful tools in geometric stability, used to resolve open problems in classification theory, in the proof of the trichotomy for Zariski geometries, and thereby the crucial component of the model theoretic solution of the function field Mordell-Lang and number field Manin-Mumford conjectures. More recently, the group configuration theorem and its avatars have taken center stage in applications to combinatorics, for example in the work of Bays-Breuillard on extensions of the Elekes-Szabó theorem. The model theoretic study of valued fields provides another example of the confluence of stability theory and algebraic model theory. A. Robinson identified ACVF, the theory of algebraically closed nontrivially valued fields, as the model companion of the theory of valued fields already in 1959, and for most of the next half century the theory maintained an “applied” character distinct from the stability theory of “pure” model theory. However, in order to describe quotients of definable sets by definable equivalence relations (imaginaries) in valued fields, Haskell, Hrushovski and Macpherson were led to the theory of stable domination and the pure and applied strands merged. The deep connections between these approaches to the theory of valued fields further manifested themselves in the Hrushovski-Loeser approach to nonarchimedian geometry, in which spaces of stably dominated types replace Berkovich spaces. Our project is structured around these three themes: First we aim to strengthen the still fairly recent relations between model theory and combinatorics. Secondly, we aim to develop the model theory of valued fields, which has traditionally been very strong both in France and in Germany, but using the sophisticated tools of geometric stability. Finally, we will develop a more abstract study of the geometric and combinatorial configurations which are a fundamental tool in the previous two subjects.

GeoMod是法国和德国的合作研究项目。现代模型理论是从一阶逻辑的角度研究数学结构的抽象性质。它试图分离可定义集合的组合属性，例如某些构型的存在性，或秩函数的存在性，并使用它们来获得结构结果。这些本质上可能是代数的或几何的，并且可以应用于特定的结构，如Berkovich空间，微分-微分代数几何，加性组合或Erdős几何。群组态定理是一个很好的例子，它表明某些组合模式必然由一个群引起，而且可能产生这种组态的群的结构是高度受限的。这个结果推广了几何代数的协调定理，赫鲁霍夫斯基在1986年给出了稳定理论的确定形式，此后成为几何稳定性中最有力的工具之一，用于解决分类理论中的开放问题，用于证明Zariski几何的三分法，从而成为函数场莫德尔-朗猜想和数场曼宁-芒福德猜想的模型理论解的关键组成部分。最近，群组态定理和它的化身在组合学的应用中占据了中心位置，例如在bayes - breuillard关于Elekes-Szabó定理扩展的工作中。数值场的模型理论研究是稳定性理论与代数模型理论相互融合的又一例证。A. Robinson早在1959年就确定了ACVF，即代数闭非平凡值场理论，作为值场理论的模型伴侣，并且在接下来的半个世纪的大部分时间里，该理论保持了与“纯”模型理论的稳定性理论不同的“应用”特征。然而，为了用值域中的可定义等价关系（虚）来描述可定义集合的商，Haskell、Hrushovski和Macpherson提出了稳定支配理论，并将纯和应用链合并。这些有值场理论的方法之间的深刻联系在非阿基姆几何的赫鲁晓夫斯基-洛泽方法中进一步表现出来，在这种方法中，稳定主导类型的空间取代了Berkovich空间。我们的项目是围绕这三个主题构建的：首先，我们的目标是加强模型理论和组合学之间最近的关系。其次，我们的目标是发展有值场的模型理论，这在法国和德国传统上都非常强大，但使用复杂的几何稳定性工具。最后，我们将对几何构型和组合构型进行更抽象的研究，这是前两门课程的基本工具。