Symmetric Spaces and Geometric Group Theory

对称空间和几何群论

• 批准号: RGPIN-2014-04521
• 负责人: Silberman, Lior
• 金额: $ 1.68万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=587857
• 关键词: Symmetric; Spaces; Geometric; Group; Theory

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. 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, leading to specific expectations when this 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, especially when the symmetries are related to number theory -- to arithmetic properties of whole numbers. In that case the techniques that others (and myself) are developing also have application to deep problems in analytic number theory. Recently I started looking at the problem for curved spaces constructed by random methods. I am also intereted in the study of curved space 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. For example, we have showed certain symmetry structures (groups) are too complicated to be the symmetry structure of a smooth space (under certain restrictions). More recently I have become interested in the topology (deformation behaviour) of symmetries. In this setting one again asks in how many ways a single symmetry structure (group) can be applied to a fixed space, and whether deformation of the symmetry structure can simplify it. I also work on the vibrational modes of random networks ("graphs"). This provides information about their connectivity and this area is a useful training ground for undergraduate research students.

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。 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）的预测通知了量子力学模型，从而导致当这种经典（台球）运动高度混乱时，具体期望。 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，我计划继续朝这个方向发展，尤其是当对称性与数字理论相关时 - 算术特性整数。 在这种情况下，他人（和我本人）正在开发的技术也对分析数理论中的深度问题进行了应用。 最近，我开始研究由随机方法构建的弯曲空间的问题。 一般而言，我也在研究弯曲空间的研究中，尤其是“扭结”的空间，这些空间可能具有角落，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）。 例如，我们显示某些对称结构（组）太复杂，无法成为光滑空间的对称结构（在某些限制下）。 最近，我对对称性的拓扑（变形行为）感兴趣。 在这种设置中，再次询问单个对称结构（组）可以应用于固定空间，以及对称结构的变形是否可以简化它。 我还在随机网络的振动模式下（“图形”）。 这提供了有关其连通性的信息，该领域是本科生研究专业学生的有用培训场。