||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 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
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.