Geometric structures in low dimensions

低维几何结构

• 批准号: RGPIN-2017-05403
• 负责人: Charette, Virginie
• 金额: $ 1.46万
• 依托单位: Université de Sherbrooke
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=765480
• 关键词: Geometric; structures; low; dimensions

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.

高等泰希穆勒理论是对由双曲平面的一组等距的离散作用量的作用所产生的高维几何结构的研究。 这是一个美丽的发展中的研究领域，丰富的贡献，从理论的希格斯束，最大表示，动力学，等之间的经典二维双曲几何和这个高维理论，丰富的有趣的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在表面上的双曲结构，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。 其次，在我看来，这一点同样重要，它们很容易被用于实验和可视化项目，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。在较低维度中，存在着“幸运的意外”，其中结构的多样性重合，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，在高维李群中变形双曲平面的等距离散群可能会产生有趣的新构造，因此，我在未来五年的计划将在某些空间的等距群中变形此类群的主题下进行，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，使本科研究的一个重要组成部分。