Relational Structures and Applications

这项研究计划旨在进一步了解融合类的Frissé理论与齐次结构、Ramsey理论和自同构群的拓扑动力学之间的联系，从而推动新的应用。我们建议作为这个项目的一部分正式加入我们的努力，特别是提出以下研究路线：**预紧展开*目标是确定齐次结构的最优展开，以获得具有扩展性质的Ramsey类，从而提供描述泛极小流的存在性和方法。这里的方法是在最少数量的精心选择的谓词的帮助下捕获基本的结构属性，这与本团队熟知的规范分区的工作非常相关。**振动稳定性*著名的扭曲问题是问希尔伯特球面是否近似不可分割。引入了拓扑群的振荡稳定性概念，证明了这些概念对齐次度量空间是等价的。随后得到了几个结果，包括证明了产生Urysohn球的振荡稳定性的一个一般结果。这是一个非常丰富的领域，非常受分析问题的影响，但有一些基本问题尚未解决。**自同构群的超群*我们已经探索了Rado图的自同构群的各种自然超群，推广了众所周知的约简方法。这项工作中令人惊讶的结果表明，进一步研究其他众所周知的齐次结构的相应的扩群的工作，与研究可数齐次结构的副本超图非常相关，从而提供了关于广义图同胚保持副本的映射的洞察。**约束识别*最近有涉及齐次结构、泛范畴和约束满足性的联系应用于复杂问题。我们建议研究泛函范畴结构的约简和上述主题的一些非常自然的问题。**区分数*这个奇怪的概念是对称性的粗略度量，我们猜想原始结构应该有2或欧米伽的区别数。这项工作似乎与无限置换群领域内的类似猜想非常相关，我们建议将这一参数的研究扩展到齐次结构。**同构：这个基本概念探索了两个关系结构嵌入另一个关系结构的条件，当一个人研究固定年龄的关系结构上的可嵌入性的前序时，自然会产生这种情况。在嵌入的背景下，集合的Cantor-Bernstein定理的失败是应用于研究线性序、图和竞赛的结果的关键。我们正在使用关系结构研究中的这些工具，例如可链接性和核的概念，以在这一领域取得进展，到目前为止，这一领域主要只针对图和树进行研究。**C-XY同质结构*这些推广了经典的同质概念。D.Lockett最近完成了对C-HH结构的分类，一个团队的NSERC USRA在有限C-MH情况下取得了实质性进展，为在可数情况下对这类结构的完整分类带来了新的希望。**关系结构中的区间*我们已经开发了一些技术，包括包含区间、区间分解和词典产品的概念，以理解原始结构和COP-WIN图。我们将在无限的背景下进一步研究临界原性的概念。

This research program aims at further understanding the connections between the Fraissé theory of amalgamation classes and homogeneous structures, Ramsey theory, and topological dynamics of automorphism groups, driven toward new applications. We propose to formally join our efforts as a team as part of this program, in particular proposing the following lines of research:**Precompact expansions*The objective is to determine an optimal expansion of homogeneous structures to obtain a Ramsey class with the expansion property, thus providing the existence and means to describe the universal minimal flows. The approach here is to capture basic structural properties with the help of a minimal number of well chosen predicates, very much related to the work on canonical partitions well known by this team. **Oscillation stability *The famous distortion problem asked whether the Hilbert sphere is approximately indivisible. The notion of oscillation stability was introduced for topological groups, and it turns out that these concepts are equivalent for homogeneous metric spaces. Several results followed, including the proof of a general result that yields the oscillation stability of the Urysohn sphere. This is a very rich area very much influenced by problems in analysis, but with some unresolved fundamental questions.**Overgroups of automorphism groups*We have explored various natural overgroups of the automorphism group of the Rado graph, generalizing well known methods for reducts. The surprising results in this work point to further work investigating the corresponding overgroups of the other well known homogeneous structures, work very much related to the study of the hypergraph of copies of countable homogeneous structures providing insight into which maps generalizing graph homeomorphisms preserve the copies.**Constraint Recognition*There are recent connections involving homogeneous structures, omega-categoricity and constraint satisfaction with applications to complexity problems. We propose to investigate some very natural questions general to the study of reducts of omega-categorical structures and the themes above.**Distinguishing number*This curious notion is a rough measure of symmetry, and we have conjectured that primitive structures should have a distinguishing number of either 2 or omega. This work appears very much related to similar conjectures within the realm of infinite permutation groups, and we propose to extend the investigation of this parameter for homogeneous structures. **Equimorphy*This fundamental notion explores conditions when two relational structures embed in the other, arising naturally when one studies the preorder of embeddability on relational structures of a fixed age. The failure of the Cantor-Bernstein theorem for sets in the context of embeddings is key to results that are applied in the study of linear orders, graphs and tournaments. We are using these tools from the study of relational structures, such as the notion of chainability and kernel to make advances in this area which has to date largely been studied only for graphs and trees.**C-XY homogeneous structures*These generalize the classical homogeneity notion. The classification of the C-HH structures was recently done by D. Lockett, and a team's NSERC USRA has made substantial progress in the finite C-MH case, providing renewed hope for the complete classification of this class in the countable case. **Intervals in relational structures*We have developed techniques, including the notion of inclusive intervals, interval decomposition and lexicographic products, toward understanding primitive structures and cop-win graphs. We will further investigate notions of critical primitivity in the infinite context.