  This thesis presents three results in geometric analysis. We first analyze
the curve-shortening flow on figure eight curves in the plane. Afterwards, we
examine the point-wise curvature preserving flow on space curves. Lastly, we
present an abridgment of our work on a family of three-dimensional Lie groups,
which, when equipped with canonical left-invariant metrics, interpolate between
Sol and hyperbolic space.
