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).