Automorphism groups of homogeneous structures

这个项目是关于模型理论(数理逻辑)、组合学和置换群理论之间的接口。有限关系语言(如图)上的可数无限结构是齐次的，如果有限子结构之间的任何同构扩展到整个结构的自同构(这大致意味着，如果两个有限块看起来相同，则整个结构的对称性将一个带到另一个)。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有有限多个‘闭’超群。在过去的15年里，由于与拓扑动力学、组合学(Ramsey理论)以及理论计算机科学中的约束满足问题的相互作用，这一猜想受到了高度关注。这个猜想仍然是开放的，但已经在一些特定的情况下得到了验证，现在似乎可以适用于广泛的结构类别。这个PHD项目的主要目的是证明Thomas关于有限刺阶NIP齐次结构的猜想-这些是当前感兴趣的基于模型理论的驯服条件，由于P.Simon最近的进展，这类已经变得可访问(同时仍然丰富)。计划中的方法是将西蒙的结构理论与博迪斯基和平斯克发展的拉姆齐理论方法相结合。最初的目标是对某些二元“圆形”结构的约简进行分类，更一般地，对NIP等级1的结构的约简进行分类。该项目还将检验某些无限刺阶的NIP结构的猜想，以期找到潜在的反例。

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.