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.

我建议在几个环境中学习几何学和动力学。许多数学问题可以解释为关于物理系统的问题；这使人们对它们的工作原理有了更多的洞察和直觉。与此相关的是，弯曲空间的“振动模式”也可以解释为量子粒子在该环境中运动的能态，并通过了解量子粒子的行为来研究。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，因此经典前量子物理学(粒子在空间中经历台球运动)的预测为量子力学模型提供了信息。这既提供了对振动模式的具体期望，也允许研究人员使用经典动力学的技术，特别是当经典(台球)运动高度混乱的时候。最近在这类问题上取得了进展，主要是在底层空间具有很大程度的对称性的情况下，我计划继续朝着这个方向工作，无论是当对称性与数论有关时--与整数的算术性质有关--或者当它们不相关时。在第一种情况下，其他人(和我)正在开发的技术也适用于解析数论中的深层问题。我还在尝试一种随机的方法来创造弯曲的空间，看看它是否能创造出具有有趣特性的空间。我对一般的曲线空间的几何学特别感兴趣，特别是可以有角、圆锥点等的“扭结”空间。有时，将想法从平坦而光滑的空间转移到这种粗糙的环境中是可能的。特别是，我研究了这种空间的对称性(或缺乏对称性)。最近的一个问题是研究对称性，这种对称性允许在某种程度上拉伸空间。我也在研究一些变形空间的行为。最后，研究了随机网络(“图”)的振动模式。这既是上面讨论的问题的测试用例，但足够有趣的是，它还提供了有关网络连接和其他属性的信息。这项研究一直是本科生研究项目的沃土。