Geometric and Combinatorial Configurations in Model Theory

GeoMod是法国和德国之间的合作研究项目。现代模型论从一阶逻辑的角度研究抽象结构的抽象性质。它试图隔离组合性质的可定义的集合，如存在某些配置，或秩函数，并使用它们来获得结构的后果。这些可以是代数或几何性质的，并且可以应用于特定的结构，如Berkovich空间，差分微分代数几何，加法组合学或Erdovich几何。一个很好的例子，配置意味着代数结构是群配置定理，它断言某些组合模式必然由群诱导，而且，可产生这种构型的基团的结构受到高度限制。这一结果推广了几何代数的坐标化定理，Hrushovski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公开问题，证明Zurski几何的可分性，从而成为函数场Mordell-Lang和数域Manin-Mumford结构的模型论解的关键组成部分。最近，群构型定理及其化身在组合学的应用中占据了中心地位，例如在贝叶斯-布罗伊亚尔关于Elekes-Szabó定理的扩展的工作中。值域的模型论研究提供了稳定性理论和代数模型论合流的另一个例子。A.罗宾逊确定ACVF，理论的代数封闭nontrivially价值领域，作为模型伴侣的理论价值领域已经在1959年，并为大多数下半个世纪的理论保持了“应用”的特点，有别于稳定理论的“纯”模型理论。然而，为了用值域中的可定义等价关系（等价关系）来描述可定义集合的等价性，Haskell、Hrushovski和Macpherson被引入了稳定控制理论，并将纯粹和应用链合并。这些方法之间的深刻联系的理论价值的领域进一步表现在赫鲁绍夫斯基-勒泽的方法，非阿基米德几何，其中空间的稳定控制类型取代伯科维奇空间。我们的项目是围绕这三个主题：首先，我们的目标是加强模型理论和组合学之间的关系仍然相当近。其次，我们的目标是发展模型理论的价值领域，这一直是非常强大的传统上在法国和德国，但使用复杂的工具的几何稳定性。最后，我们将发展一个更抽象的研究的几何和组合的配置，这是一个基本的工具，在前两个主题。

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.