Geometric structures in low dimensions

高阶Teichmüler理论是研究双曲平面上一组等距线的离散作用所产生的高维几何结构。这是一个发展迅速的研究领域，得到了希格斯丛、极大表示、动力学等理论的丰富。在经典的二维双曲几何和这个高维理论之间，有大量有趣和有些具体的例子。*在过去的五年里，我的研究重点是从曲面上的双曲结构产生的几何结构，三维和四维。具体地说，我研究了仿射洛伦兹空间上的适当群作用及其共形紧化，以及双圆盘。研究三维和四维的几何结构很重要，原因有两个。首先，低维的例子为高维的情况提供了有价值的见解，特别是在高阶Teichmüler理论中。其次，在我看来，这一点同样重要，他们很容易参与实验和可视化项目，这反过来又为年轻人提供了一个研究的切入点，特别是本科生和硕士学生。*在较低的维度，有多种结构重合的“快乐意外”，允许用不同的语言重新表述观察和问题。我为之做出贡献的仿射洛伦兹思想可以推广到更广泛的背景下。例如，通过变形高维李群中双曲平面的离散等距线群，可能会产生有趣的新构造。*因此，我未来五年的计划将在某些空间的等距线群中变形这些群的主题下进行，并研究出现的几何结构。我将特别强调可视化和计算机实验，这将使本科生的研究成为重头戏。

Higher Teichmüller theory is the study of higher dimensional geometric structures arising from the action of a discrete action of a group of isometries of the hyperbolic plane. It is a beautifully developing area of research, enriched by contributions from the theory of Higgs bundles, maximal representations, dynamics, etc. Between classical two-dimensional hyperbolic geometry and this higher dimensional theory, a wealth of interesting and somewhat concrete examples abound.******Over the past five years, my research focused on geometric structures emerging from hyperbolic structures on surfaces, in dimensions three and four. Specifically, I have examined proper group actions on affine Lorentzian three-space and its conformal compactification, as well as the bidisk. Geometric structures in dimensions three and four are important to study for two reasons. First, lower-dimensional examples provide valuable insights for higher dimensional cases, especially in higher Teichmüller theory. Second, and in my view this is just as important, they readily lend themselves to experimentation and visualisation projects, which in turn offer an entry point into research for younger people, especially undergraduate and Master's students.******In lower dimensions, there are ``happy accidents'' where a diversity of structures coincide, allowing observations and questions to be reformulated in a different language. Affine Lorentzian ideas, to which I have contributed, could be generalized to a wider context. For example, interesting new constructions may result from deforming discrete groups of isometries of the hyperbolic plane in higher dimensional Lie groups.******Therefore, my program for the next five years will be pursued under the theme of deforming such groups in isometry groups of certain spaces, and studying the geometric structures that arise. I will place a particular emphasis on visualisation and computer experimentation, enabling a heavy component in undergraduate research.