On Flows in Teichmüller and Moduli Spaces of Surfaces

Herein we investigate compactifcations of both the Teichmüller and moduli spaces of surfaces with boundary. Using the pair of pants, the one-holed torus, and 4-holed sphere as motivating examples, we explore the notion of owing to the boundaries of the Teichmüller and moduli spaces by moving the boundary curves on the underlying surface. We begin by, in each case, defining the degenerate objects which comprise the boundary. We then explore and compute exactly what these flows must be. In the cases of the pair of pants and the one-holed torus, we give an explicit flow. In the case of the 4-holed sphere, we compute several obstructions to this flow, each dependent on a choice of symmetry. We then construct the degenerate objects for the general case, followed by a different set explicit of flows, each due to a choice of decomposing curves