Geometric structures in low dimensions

高等泰希米勒理论是研究由双曲平面的一组等距的离散作用所产生的高维几何结构。 这是一个美丽的发展领域的研究，丰富的贡献，从理论的希格斯束，最大代表性，动力学等之间的经典二维双曲几何和这个高维理论，丰富的有趣和有点具体的例子比比皆是。 在过去的五年里，我的研究集中在几何结构出现的双曲结构的表面上，在三维和四维。 具体地说，我研究了仿射洛伦兹三维空间上的真群作用及其共形紧化，以及双圆盘。 三维和四维空间的几何结构研究很重要，原因有二。 首先，低维的例子提供了有价值的见解高维情况下，特别是在较高的Teichmüller理论。 第二，在我看来，这也同样重要，他们很容易把自己借给实验和可视化项目，这反过来又为年轻人，特别是本科生和硕士生提供了一个进入研究的切入点。 在较低的维度中，存在着“幸运的意外”，其中各种各样的结构重合，允许观察和问题以不同的语言重新表述。 仿射洛仑兹思想，我已经作出了贡献，可以推广到更广泛的背景。 例如，有趣的新结构可能会导致变形的离散群的等距的双曲平面在高维李群。 因此，我在未来五年的计划将在变形的主题下进行，这些群体在某些空间的等距群，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，使本科研究的一个重要组成部分。

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.