Title: A comparison of the Ricci ﬂow and related constructions on Riemann surfaces for ﬁnding canonical metrics. Description: This project will investigate some Riemann surfaces --- i.e. those surfaces which can locally be described by a complex coordinate --- and in particular some well known constructions to ﬁnd what are called canonical metrics on them. These are, in a precise sense, the most natural metrics one can place on these surfaces. Among these well known constructions are the Ricci ﬂow, which is a PDE introduced by Richard Hamilton that can be prescribed on the surface, as well a construction introduced by Simon Donaldson called a T-iteration. This project will introduce these ideas to summer research students and will seek to make a comparison of their effectiveness. Background: Familiarity with complex analysis (408) is a must. Further helpful courses as background would be: Complex II (409), PDEs (418), Topology (440 or 540), Differential Geometry (404), Linear Algebra II and III (306, 406).