Symmetric Spaces and Geometric Group Theory

点击翻译按钮获取中文摘要

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.