Geometric and Combinatorial Configurations in Model Theory

GeoMod是法国和德国之间的一个合作研究项目。当代模型理论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性质，例如某些配置的存在，或秩函数的存在，并使用它们来获得结构结果。这些结构本质上可以是代数的或几何的，可以应用于特定的结构，如Berkovich空间、差分-微分代数几何、可加组合学或ErdőS几何。蕴含代数结构的配置的一个很好的例子是群配置定理，它断言某些组合模式必然由一个群诱导，而且可能产生这种配置的群的结构受到高度限制。这个推广了几何代数协调定理的结果，在1986年由Hrushovski给出了稳定理论的明确形式，此后成为几何稳定的最有力的工具之一，用于解决分类理论中的公开问题，用于证明Zariski几何的三分性，从而成为函数域Mordell-Lang和数域Manin-Mumford猜想模型理论解的关键组成部分。最近，群组态定理及其化身在组合学的应用中占据了中心舞台，例如在Bays-Breuillard对Elekes-Szabó定理的扩展工作中。值域的模型论研究是稳定性理论和代数模型理论融合的又一例证。A·罗宾逊早在1959年就将代数封闭的非平凡值域的ACVF理论确定为值域理论的模型伴侣，在接下来的半个世纪的大部分时间里，该理论保持着有别于“纯”模型理论的稳定性理论的“实用”性质。然而，为了用值域上的可定义等价关系(想象)来描述可定义集合的商，Haskell、Hrushovski和Macpherson被引入了稳定支配理论，并将纯的和应用的链合并。这些方法与值域理论之间的深层联系进一步表现在非阿基米德几何学的Hrushovski-Loeser方法中，其中稳定支配类型的空间取代了Berkovich空间。我们的项目围绕这三个主题构建：第一，我们的目标是加强模型理论和组合学之间最近才建立起来的关系。其次，我们的目标是发展值域模型理论，这一理论在法国和德国传统上都非常强大，但使用了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，这些构型是前两门课程中的基本工具。

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.