UC Santa Barbara

We study two separate systems each of which emphasizes the geometrical and topological aspects of soft condensed matter systems. The geometry side of condensed matter is exemplified by the geometrical frustration experienced by constrained thermalized membranes. We study the dynamics of the novel tilted phase of thermalized cantilevers and find that the geometry of the system plays an important role in determining the behavior of the underdamped dynamics. We then delve into the topology of soft matter by studying the non-Abelian braiding of singular defect lines of systems with biaxial symmetry. Biaxial nematic defects have a non-Abelian topology that allows for the formation of topologically stable braided structures. We devise a braid theory that incorporates strand labeling via colors and allows for crossing relations that take colorings into account. We use this colored braid theory to translate complex braided structures into algebraic expressions that can be used to determine whether the braid is entangled under the algebra of its assigned fundamental group. We supplement this with possible experimental realizations of the non-Abelian structures inherent to the biaxial nematic system.