Interactions between dynamics, topology and combinatorics in low dimensions

低维动力学、拓扑学和组合学之间的相互作用

• 批准号: 2749484
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2749484
• 关键词: Interactions; between; dynamics; topology; combinatorics; Interactions; between; dynamics; topology; combinatorics

There is a history of rich interplay between chaotic dynamical systems and the topological properties of the underlying space. When the dynamical system is a flow, these relationships date back to classical work on so-called hyperbolic or Anosov flows.In dimension three, a modern rendition of a hyperbolic flow is a pseudo-Anosov flow. These flows have become important in the theory of hyperbolic three-manifolds due to the work, for example, of Fried, Calegari, Christy, Fenley and Mosher. Some such deep relationships involve the dynamics of surface diffeomorphisms, homotopic properties of periodic orbits of the flow, and the large-scale geometry of the underlying manifold. Among recent results on pseudo-Anosov flows is a remarkable correspondence between pseudo-Anosov flows veering triangulations. Veering triangulations are a purely combinatorial structure introduced by Agol. In addition to dynamics, the triangulations illuminate various topological properties of the underlying space.We will further explore the relationships between pseudo-Anosov flows and the theory of three-manifolds. A primary motivation will be the correspondence with veering triangulations. Trading a flow for a veering triangulation allows us to study the flow with combinatorics and computation. Some problems motivating our work are deciding when an arbitrary flow admits a veering triangulation, determining when two flows are equivalent, enumerating all flows, and understanding how modifying the flow modifies its triangulation. Such problems will require novel techniques and a detailed understanding of the current machinery.

混乱的动力系统与基础空间的拓扑特性之间存在丰富的相互作用的历史。当动态系统是流动的流动时，这些关系可以追溯到所谓的双曲线或Anosov流的经典作品。在第三维中，双曲线流的现代化是伪-Anosov流。由于炸，Calegari，Christy，Fenley和Mosher的作品，这些流动在双曲线三序列理论中变得很重要。某些深厚的关系涉及表面差异性的动力学，流动周期轨道的同型特性以及基础歧管的大规模几何形状。关于伪anosov流的最新结果包括伪anosov流动三角剖分之间的显着对应关系。旋转三角形是Agol引入的纯粹组合结构。除动态外，三角形阐明了基础空间的各种拓扑特性。我们将进一步探索伪-Anosov流与三个序列理论之间的关系。主要的动机将是与三角剖分的对应关系。将流量交换以进行旋转三角剖分，使我们能够通过组合和计算来研究流程。促使我们工作的一些问题正在决定何时任意流动的三角剖分，确定何时两个流相同，列举所有流量，并了解修改流程如何修改其三角测量。这样的问题将需要新颖的技术以及对当前机械的详细理解。