Parabolic flows in geometry

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The PI proposes to study various problems in the field of geometric partial differential equations. For example, he would like to study the qualitative behavior of the Ricci flow on manifolds with positive isotropic curvature. In particular, the PI would like to analyze what kinds of singularities can occur along the flow. In another project, the PI intends to study the Yamabe problem on manifolds with boundary. The goal here is to construct conformal metrics that have constant scalar curvature in the interior and zero mean curvature along the boundary. The PI also intendsto study ancient solutions to the Yamabe flow on the n-dimensional sphere. This question is motivated by a classification result, due to Hamilton, Daskalopoulos, and Sesum, for ancient solutions to the Ricci flow on the two-sphere. This project is concerned with questions at the crossroads of analysis and differential geometry. The use of analytical techniques in geometry has been extremely successful in recent years. In particular, Hamilton's Ricci flow plays a key role in Perelman's solution of the Poincare conjecture, as well as in the proof of the Differentiable Sphere Theorem by Richard Schoen and myself. The goal of the project is to gain a better understanding of these geometric evolution equations, and their qualitative properties.