Interactions between dynamics, topology and combinatorics in low dimensions

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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.