Point-Vortex Dynamics on Non-Orientable and Multiply-Connected Surfaces

This thesis explores geometric aspects of point-vortex dynamics on surfaces. We review background material in geometric mechanics and Hodge theory, and discuss the geometric view of ideal hydrodynamics as an infinite-dimensional Lie-Poisson system. We then explore the finite-dimensional Hamiltonian system of point-vortex dynamics within this framework. We begin by studying the dynamics of point-vortices on simply-connected, orientable surfaces. We then explore two more convoluted cases: non-orientable and multiply-connected surfaces. In each case, we utilize the geometric perspective to formalize a Hamiltonian description of the point-vortex dynamics.