Symmetric Spaces and Geometric Group Theory

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。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）的预测告知量子力学模型，导致这种经典（台球）运动高度混乱时的特定预期。最近在这些问题上取得了进展，主要是在底层空间具有很大程度对称性的情况下，我计划继续在这个方向上工作，特别是当对称性与数论有关时-整数的算术性质。在这种情况下，其他人（和我）正在开发的技术也可以应用于分析数论中的深层问题。最近我开始研究由随机方法构造的弯曲空间的问题。我也对一般弯曲空间的研究感兴趣，特别是“扭结”空间，它可以有角，锥点等。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）。例如，我们已经证明了某些对称结构（群）太复杂而不能成为光滑空间的对称结构（在某些限制下）。最近，我对对称的拓扑（变形行为）产生了兴趣。在这种情况下，人们又会问，一个对称结构（群）有多少种方法可以应用于一个固定的空间，以及对称结构的变形是否可以简化它。我也研究随机网络的振动模式（“图”）。这提供了关于它们的连接的信息，这个区域是本科生研究的一个有用的训练基地。

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.