Finitely Bounded Homogeneous Structures

齐次结构在模型论中扮演着重要的角色，它们提供了丰富的例子和反例。它们有一个很大的自同构群，可以使用无限置换群理论来研究。它们自然地出现在数学的几个领域。齐次结构的许多重要例子都是有界的，也就是说，可以用许多禁止的有限子结构来描述。通过这种方式，我们可以通过计算来存储和操作同质结构，并且关于同质结构的许多基本问题都可以作为算法问题提出。事实上，齐次结构在理论计算机科学中有应用，例如用于研究约束满足问题（CSP）的计算复杂性，用于研究关系代数的正规表示，或可定义结构的计算。结构拉姆齐理论是研究均匀结构的一个相对年轻而有力的工具。齐次结构的拉姆齐定理在上述应用领域中有应用，但也在拓扑动力学中有应用，例如用于验证拓扑群的极端顺从性，顺从性和唯一遍历性。 该项目的目标之一是回答关于大类同质结构的基本开放问题，例如，通过对轨道增长率施加进一步的模型理论假设或限制而获得的类。通过这种方式，我们也希望获得上述应用领域的见解。

Homogeneous structures play an important role in model theory, where they provide a rich source of examples and counterexamples. They have a large automorphism group and can be studied using the theory of infinite permutation groups. They naturally appear in several areas of mathematics. Many of the important examples of homogeneous structures are *finitely bounded*, that is, can be described by finitely many forbidden finite substructures. In this way we store and manipulate homogeneous structures computationally, and many fundamental questions about homogenous structures can be posed as algorithmic questions. Indeed, homogeneous structures have applications in theoretical computer science, for example for the study of the computational complexity of constraint satisfaction problems (CSPs), for the studying normal representations of relation algebras, or computation with definable structures. A relatively young and powerful tool in the study of homogeneous structures is structural Ramsey theory. Ramsey theorems for homogeneous structures have applications in the mentioned application areas, but also in topological dynamics, for example for verifying extreme amenability, amenability, and unique ergodicity of topological groups. One of the goals of the project is to answer fundamental open questions about homogeneous structures for large classes, e.g., classes obtained by imposing further model-theoretic assumptions or restrictions on the orbit growth rate. In this way we hope to obtain insights for the mentioned application areas as well.