Geometric structures in low dimensions

高泰希<e:1>勒理论是研究由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的理论。这是一个发展良好的研究领域，希格斯束理论、极大表示、动力学等的贡献丰富了这一领域。在经典的二维双曲几何和这种高维理论之间，有大量有趣而又有些具体的例子。******在过去的五年中，我的研究集中在三维和四维曲面上的双曲结构中出现的几何结构。具体地说，我研究了仿射洛伦兹三维空间上的适当群作用及其共形紧化，以及双盘。三维和四维几何结构的研究很重要，原因有二。首先，低维的例子为高维的情况提供了有价值的见解，特别是在更高的teichm<s:1> ller理论中。其次，在我看来，同样重要的是，他们很容易参与实验和可视化项目，这反过来又为年轻人，尤其是本科生和硕士生提供了一个研究的切入点。******在较低的维度中，有“快乐的意外”，多种结构重合，允许用不同的语言重新表述观察和问题。我提出的仿射洛伦兹思想，可以推广到更广泛的背景下。例如，在高维李群中，双曲平面的等距离散群的变形可能会产生有趣的新结构。******因此，我未来五年的计划将以在特定空间的等距群中变形这样的群为主题，并研究由此产生的几何结构。我将特别强调可视化和计算机实验，使其成为本科研究的重要组成部分。

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.