Henry Liu, Department of Pure Mathematics, University of Waterloo
Now that we have some understanding of how the moduli space of Riemann surfaces arises from the Teichmller space, we can do geometry on it. We will motivate why it is useful to compactify moduli spaces and explain why the moduli space we have so far is not compact. The Deligne–Mumford compactification arises from adding Riemann surfaces with nodes into the moduli space. We will state and prove Mumford’s compactness criterion, which gives clues as to why adding such points into the moduli space is enough to compactify it.