Symmetric spaces, topology and analysis

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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.