Symmetric spaces, topology and analysis

我建议在几种情况下学习几何和动力学。许多数学问题可以解释为物理系统的问题；这为他们的工作提供了额外的洞察力和直觉。与此相关的是，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，并通过理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）的预测可以为量子力学模型提供信息。这既提供了关于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）运动高度混乱时。最近在这类问题上已经取得了进展，主要是在底层空间具有很大程度对称性的情况下，我计划在这个方向上继续我的工作，无论是在对称性与数论有关的情况下——与整数的算术性质有关的情况下——还是在它们不是的情况下。在第一种情况下，其他人（和我）正在开发的技术也可以应用于解析数论中的深层问题。我也在尝试一种创建弯曲空间的随机方法，看看它是否可以创建具有有趣属性的空间。一般来说，我对弯曲空间的几何形状很感兴趣，尤其是那些有角、锥点等的“扭结”空间。有时可以将想法从平坦和光滑的空间转移到这种粗糙的环境中。特别是，我研究这些空间的对称性（或缺乏对称性）。最近的一个问题是研究允许在一定程度上拉伸空间的对称性。我也在研究一些空间在变形下的行为。最后，我研究了随机网络（“图”）的振动模式。这既是上面讨论的问题的一个测试用例，但有趣的是，它还提供了有关网络的连通性和其他属性的信息。这项研究为本科生的研究项目提供了肥沃的土壤。

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.