Automorphism groups of homogeneous structures

齐次结构的自同构群

• 批准号: 2712596
• 金额: --
• 依托单位: University of Leeds
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2712596
• 关键词: Automorphism; groups; homogeneous; structures

This project is on the interface between model theory (mathematical logic), combinatorics, and permutation group theory. A countably infinite structure over a finite relational language (such as a graph) is homogeneous if any isomorphism between finite substructures extends to an automorphism of the whole structure (this means, roughly, that if two finite pieces look the same, then there is a symmetry of the whole structure taking one to the other). S. Thomas conjectured in 1996 that every such structure has finitely many `reducts', that is, the automorphism group has finitely many `closed' supergroups in the full symmetric group. This conjecture has received high attention over the last 15 years due to interactions with topological dynamics, with combinatorics (Ramsey theory), and with constraint satisfaction problems in theoretical computer science. The conjecture remains wide open but has been verified in some specific cases, and now seems accessible for wide classes of structures. The main aim of this PhD project is to prove Thomas's conjecture for NIP homogeneous structures of finite thorn rank - these are model-theoretic tameness conditions of high current interest, and the class has been made accessible (whilst still being rich) due to recent advances of P. Simon. The planned approach is to combine Simon's structure theory with a Ramsey-theoretic method developed by Bodirsky and Pinsker. Initial goals are to classify the reducts of certain binary `circular' structures, and more generally of NIP rank 1 structures. The project will also examine the conjecture for certain NIP structures of infinite thorn rank, with a view to potential counterexamples.

该项目是模型论（数理逻辑）、组合学和置换群理论之间的接口。如果有限子结构之间的任何同构延伸到整个结构的自同构，则有限关系语言（例如图）上的可数无限结构是同构的（这大致意味着，如果两个有限部分看起来相同，则存在整个结构的对称性）。 S. Thomas在1996年猜想，每个这样的结构都有有限多个“约简”，即自同构群在完全对称群中具有有限多个“闭”超群。由于与拓扑动力学、组合学（拉姆齐理论）以及理论计算机科学中的约束满足问题的相互作用，该猜想在过去 15 年中受到了高度关注。该猜想仍然是开放的，但已在某些特定情况下得到验证，现在似乎可用于广泛的结构类别。 这个博士项目的主要目的是证明托马斯关于有限刺秩的 NIP 齐次结构的猜想 - 这些是当前高度关注的模型理论驯服条件，并且由于最近的进展，该类已经变得易于使用（同时仍然很丰富） P.西蒙.计划的方法是将西蒙的结构理论与博德斯基和平斯克开发的拉姆齐理论方法相结合。最初的目标是对某些二元“循环”结构以及更一般的 NIP 1 级结构的约简进行分类。该项目还将检验某些无限刺秩 NIP 结构的猜想，以期找到潜在的反例。