Geometry and computability in low-dimensional topology and group theory.

低维拓扑和群论中的几何和可计算性。

• 批准号: 2283616
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2283616
• 关键词: Geometry; computability; low; dimensional; topology

With the proof of Thurston's Geometrisation Conjecture, 3-manifolds are in some sense classified. However, the non-constructive nature of the proof gives no direct method for deciding whether two given 3-manifolds are the same. In this project, Ascari will examine these computational questions. He will approach it using several techniques. On the one hand, topological tools such as hierarchies will prove to be useful. On the other hand, algebraic approaches, such as the use of the profinite completion of the fundamental group, will also be fruitful. Overlying both these approaches is the pervasive role of hyperbolic geometry, which is an area of expertise of Ascari. A field where these questions are particularly of interest is knot theory, which Ascari will also examine. We now explain these approaches in a bit more detail. The profinite completion of a finite generated group is a compact topological group that is the inverse limit of its finite quotients. It is conjectured that the profinite completion of the fundamental group of a hyperbolic 3-manifold should completely determine the manifold. If true, this would provide a new solution to the homeomorphism problem for 3-manifolds. Both of Dario's supervisors have expertise in this area. Lackenby has used profinite completions to study finite-sheeted covers of 3-manifolds, and Bridson has recently discovered the first example of a hyperbolic 3-manifold that is determined by its profinite completion. In fact, it is determined by its profinite completion among all finitely generated residually finite groups. A hierarchy for a 3-manifold is a finite sequence of decompositions along incompressible surfaces that cuts the manifold into 3-balls. Haken showed that many 3-manifolds have a hierarchy; in particular, all knot complements have one. He used these to produce the first solution to the equivalence problem for knots and links. Lackenby has used hierarchies to produce quantitative bounds on the computational complexity of this and related problems. For example, he showed that the problem of recognising the unknot lies in the complexity class co-NP. In his project, Dario will use these methods to analyse more complicated knots. This will tie in with the study of profinite completions, since it seems likely that a proof of profinite rigidity will need to use incompressible surfaces in some way. This project will draw on many different areas of expertise of Dario's supervisors Lackenby and Bridson. It will require sophisticated methods in low-dimensional topology, hyperbolic geometry and geometric group theory. The project lies in the EPSCR Research Areas Geometry & Topology and Algebra.

有了瑟斯顿的几何化猜想的证明，在某种意义上，3个manifords被分类。但是，证明的非结构性性质没有直接决定决定两个给定的3个策略是否相同。在这个项目中，Ascari将研究这些计算问题。他将使用几种技术对其进行处理。一方面，诸如层次结构之类的拓扑工具将被证明是有用的。另一方面，代数方法，例如使用基本组的涂鸦完成，也将是富有成果的。这两种方法都在上面是双曲几何形状的普遍作用，这是Ascari的专业知识领域。这些问题特别感兴趣的领域是结理论，阿卡里也将研究。现在，我们更详细地解释了这些方法。有限生成的组的仔细完成是一个紧凑的拓扑组，是其有限商的逆极限。据推测，双曲线3个manifold的基本群体的详细完成应完全确定歧管。如果是真的，这将为3个manifolds的同构问题提供新的解决方案。达里奥（Dario）的两位主管在这方面都有专业知识。拉克森比（Lackenby）使用涂鸦完成研究了3个manifolds的有限层盖子，而布里森（Bridson）最近发现了一个由夸张的3个manifold的示例，该例子由其涂鸦完成确定。实际上，它取决于其在所有有限生成的残留群体中的大量完成。 3个manifold的层次结构是沿着不可压缩的表面的分解序列，将歧管切成3球。哈肯表明，许多3个曼尼夫人具有层次结构。特别是，所有结都有一个。他用这些为结和链接的等效问题提供了第一个解决方案。 Lackenby使用层次结构在该问题和相关问题的计算复杂性上产生定量界限。例如，他表明识别未结的问题在于复杂性类别的共同体。在他的项目中，达里奥（Dario）将使用这些方法来分析更复杂的结。这将与研究的研究有关，因为似乎需要以某种方式使用不可压缩的表面证明。该项目将借鉴达里奥（Dario）主管Lackenby和Bridson的许多不同领域。它将需要低维拓扑，双曲线几何和几何组理论的复杂方法。该项目在于EPSCR研究领域的几何和拓扑以及代数。