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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711629
• 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.

较高的Teichmüller理论是研究由双曲线平面的一组异构体的离散作用作用引起的更高维几何结构。这是一个发展良好的研究领域，由希格斯捆绑理论，最大代表性，动态等的贡献丰富，在经典的二维双曲几何形状和这种更高的维度理论，丰富的兴趣和一些具体示例之间。 在过去的五年中，我的研究集中于从表面上的双曲线结构（在三个和四个方面）出现的几何结构。具体而言，我已经检查了对洛伦兹仿射三个空间及其保形的紧凑型的适当小组动作以及双风。维度三和四的几何结构对于研究很重要，原因有两个。首先，较低的示例为更高维度的情况提供了宝贵的见解，尤其是在更高的Teichmüller理论中。其次，在我看来，这同样重要，它们很容易地将自己用于实验和可视化项目，这反过来又为年轻人，尤其是本科生和硕士学生提供了研究。 在较低的维度中，有``快乐的事故''，其中各种结构重合，可以用不同的语言对观察和问题进行改革。我为之贡献的Affine Lorentzian想法可以推广到更广泛的背景。例如，有趣的新结构可能是由于在较高尺寸的谎言组中畸形双曲线平面异构体的离散组而产生的。 因此，将在未来五年的计划中以某些空间等轴测分组的形式变形，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，从而使本科研究中的重分组成。