The ninth Takagi Lectures June 4, 2011 (Sat) 10:05--11:05, 14:00
--15:00 Lecture Hall (Room No. 420) Research Institute for
Mathematical Sciences Kyoto University, Kyoto, Japan  Evolution
equations in Riemannian geometry Simon Brendle (Stanford
University)   A fundamental question in Riemannian geometry is
to find canonical metrics on a given smooth manifold. In the
1980s, R. Hamilton proposed an approach to this question based
on parabolic partial differential equations. The goal is to
start from a given initial metric and deform it to a canonical
metric by means of an evolution equation. There are various
natural evolution equations for Riemannian metrics, including
the Ricci flow and the conformal Yamabe flow. We will discuss
the global behavior of the solutions to these equations. In
particular, we will describe how these techniques can be used to
prove the Differentiable Sphere Theorem.  References: S.
Brendle, Evolution equations in Riemannian geometry, Japanese
Journal of Mathematics, Volume 6-1, (2011)