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.

通过瑟斯顿几何化猜想的证明，3-流形在某种意义上被分类了。然而，该证明的非构造性性质没有给出直接方法来确定两个给定的 3-流形是否相同。在这个项目中，阿斯卡里将研究这些计算问题。他将使用多种技术来实现这一目标。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，例如使用基本群的有限完备性，也将富有成效。覆盖这两种方法的是双曲几何的普遍作用，这是阿斯卡里的专业领域。这些问题特别令人感兴趣的一个领域是纽结理论，阿斯卡里也将研究这个领域。我们现在更详细地解释这些方法。有限生成群的有限完备性是一个紧拓扑群，是其有限商的逆极限。据推测，双曲3-流形的基本群的有限完备性应该完全确定流形。如果为真，这将为 3 流形的同胚问题提供一个新的解决方案。达里奥的两位主管都拥有该领域的专业知识。 Lackenby 使用有限完成来研究 3 流形的有限片覆盖，而 Bridson 最近发现了第一个由有限完成确定的双曲 3 流形示例。事实上，它是由它在所有有限生成的残差有限群中的有限完备性决定的。 3 流形的层次结构是沿着不可压缩表面的有限分解序列，将流形切割成 3 球。 Haken 表明许多 3 流形都具有层次结构；特别是，所有的结补品都有一个。他利用这些提出了结和链节等价问题的第一个解决方案。 Lackenby 使用层次结构对这个问题和相关问题的计算复杂性进行了定量限制。例如，他表明识别未知结的问题在于复杂性类co-NP。在他的项目中，达里奥将使用这些方法来分析更复杂的结。这将与有限完成的研究联系起来，因为有限刚性的证明似乎需要以某种方式使用不可压缩表面。该项目将利用达里奥的主管 Lackenby 和 Bridson 的许多不同领域的专业知识。这将需要低维拓扑、双曲几何和几何群论方面的复杂方法。该项目属于 EPSCR 研究领域几何、拓扑和代数。