We study the interplay of topology and geometry with chirality for several passive and active systems, employing both analytical and numerical methods. In chapter 1, we explain how nematic liquid... Show moreWe study the interplay of topology and geometry with chirality for several passive and active systems, employing both analytical and numerical methods. In chapter 1, we explain how nematic liquid crystals confined in toroidal geometries undergo structural phase transitions depending on the slenderness of the confining toroid. In chapter 2, we consider a system of active polar swimmers that align with their neighbors. When confined in the right geometry, the system will self-assemble into a state with topologically protected chiral acoustic modes. The chirality in this system manifests itself as a temporal one, rather than a spatial chirality. Chapter 3 shows how systems of Yukawa charged active spinning dimers self-assemble into a crystal phase with spatiotemporal order, a liquid phase or a glass phase depending on the density. Depending on the phase and the confinement geometry of these systems of actively spinning dimers, the system will allow for rigid body rotations or edge currents. Finally, in chapter 4 we introduce a novel method of doing molecular dynamics on curved surfaces by developing a symplectic integrator. We present preliminary results on two-dimensional crystal melting in the presence of curvature. We find that the crystal may melt inhomogeneously. Show less
