Symmetric spaces, topology and analysis

对称空间、拓扑和分析

• 批准号: RGPIN-2019-03964
• 负责人: Silberman, Lior
• 金额: $ 1.53万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737282
• 关键词: Symmetric; spaces; topology; analysis

I propose to study geometry and dynamics in several settings. Many mathematical problems can be interpreted as problems about physical systems; this gives additional insight and intuition into their workings. Relevant here, the "vibrational modes" of curved space can also be interpreted as the energy states of a quantum particle moving in that environment and studied by understanding the behaviour of quantum particles. In that case, high-frequency vibration roughly corresponds to a situation where quantum effects should be small, so that predictions from classical pre-quantum physics (in which the particle undergoes billiard motion in the space) inform the quantum-mechanical model. This both provides specific expectations about the vibrational modes and allows researchers to use techniques from classical dynamics, especially when the classical (billiard) motion is highly chaotic. There has been recent progress on such problems, mainly in the case where the underlying space has a large degree of symmetry, and I plan to continue my work in that direction, both when the symmetries are related to number theory -- to arithmetic properties of whole numbers -- and when they are not. In the first case the techniques that others (and myself) are developing also have application to deep problems in analytic number theory. I am also trying a randomized method for creating curved spaces to see if it can create spaces with interesting properties. I am separately interested in the geometry of curved spaces in general, especially "kinked" spaces which can have corners, cone points, and the like. Sometimes it is possible to transfer ideas from flat and smooth spaces to this rougher setting. In particular, I study the symmetry (or lack thereof) of such spaces. A recent problem is studying symmetries which are allowed to stretch the space to some extent. I am also working on the behaviour of some spaces under deformation. Finally, I study the vibrational modes of random networks ("graphs"). This is both a test case for the problems discussed above, but interestingly enough also provides information about the connectivity and other properties of the network. This study has been a fertile ground for undergraduate research projects.

我建议在几种设置中研究几何和动力学。许多数学问题可以解释为关于物理系统的问题。这为他们的运作提供了更多的见识和直觉。在这里相关，弯曲空间的“振动模式”也可以解释为在该环境中移动的量子粒子的能量状态，并通过了解量子颗粒的行为来研究。在这种情况下，高频振动大致对应于量子效应应该很小的情况，以便从经典的量子前物理学（粒子在空间中经历台球运动）的预测通知量子力学模型。这两者都提供了有关振动模式的特定期望，并允许研究人员使用经典动力学的技术，尤其是当经典（台球）运动高度混乱时。最近在此类问题上取得了进展，主要是在基础空间具有很大程度的对称性的情况下，我计划在与数字理论相关时继续朝着这个方向朝着这个方向发展 - 算术特性整数 - 当他们不是时。在第一种情况下，他人（和我本人）正在开发的技术也适用于分析数理论中的深层问题。我还尝试了一种随机方法来创建弯曲空间，以查看它是否可以创建具有有趣属性的空间。我对一般的弯曲空间的几何形状分别感兴趣，尤其是“扭结”的空间，这些空间可以具有角落，圆锥点等。有时，可以将想法从平坦和光滑的空间转移到更粗糙的环境。特别是，我研究了此类空间的对称性（或缺乏对称性）。最近的一个问题是研究对称性，这些对称性被允许在一定程度上扩展空间。我还在研究变形的某些空间的行为。最后，我研究随机网络的振动模式（“图形”）。这既是上述问题的测试案例，但有趣的是，还提供了有关网络连接性和其他属性的信息。这项研究是本科研究项目的肥沃基础。