Relational Structures and Applications

关系结构和应用

• 批准号: RGPGP-2014-00062
• 负责人: Laflamme, Claude
• 金额: $ 0.8万
• 依托单位: University of Calgary
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Group
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=587280
• 关键词: Relational; Structures; Applications

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.

本研究计划旨在进一步了解合并类和齐次结构，拉姆齐理论和自同构群的拓扑动力学的弗雷斯理论之间的联系，朝着新的应用驱动。我们建议正式加入我们的努力，作为这个计划的一部分，特别是提出以下研究路线： 准紧展开式 目的是确定齐次结构的最优展开，得到具有展开性质的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，这与这个团队所熟知的规范划分的工作非常相关。 振荡稳定性 著名的扭曲问题是关于希尔伯特球面是否近似不可分的。在拓扑群中引入了振荡稳定性的概念，证明了这些概念在齐次度量空间中是等价的。随后的几个结果，包括证明的一般结果，产生振荡稳定的Urysohn球。这是一个非常丰富的领域，在很大程度上受到分析问题的影响，但也有一些未解决的基本问题。 自同构群的扩群 我们已经探索了各种自然的自同构群的Rado图，推广众所周知的方法约简。 令人惊讶的结果，在这项工作中指出，进一步的工作调查相应的overgroups的其他众所周知的同质结构，工作非常相关的研究超图的副本的可数同质结构提供洞察到地图推广图同胚保持副本。 约束识别 最近，同质结构、omega-categoricity和约束满足与复杂性问题的应用之间存在联系。我们提出了一些非常自然的问题，一般的研究欧米加范畴结构和上述主题的约简。 可区别数 这个奇怪的概念是对称性的一个粗略的度量，我们已经证明原始结构应该有一个区别数2或Ω。这项工作似乎非常相关的领域内的无限置换群的类似acetrutures，我们建议扩大这一参数的均匀结构的调查。 等形性 这个基本概念探讨了两个关系结构嵌入另一个关系结构的条件，当人们研究固定年龄的关系结构的可嵌入性的前序时，自然会出现这种情况。 Cantor-Bernstein定理在嵌入背景下的失败是应用于线性序、图和竞赛图研究结果的关键。 我们正在使用这些工具，从关系结构的研究，如概念的chainability和内核取得进展，在这一领域，迄今为止，主要是研究图和树。 C-XY均匀结构 这些概括了经典的同质性概念。最近D. Lockett和一个团队的NSERC USRA在有限C-MH情况下取得了实质性进展，为可数情况下该类的完整分类提供了新的希望。 关系结构中的区间 我们已经开发了技术，包括概念的包容性区间，区间分解和词典产品，对理解原始结构和cop-win图。我们将进一步研究无限语境中的临界连续性概念。