Applications of Riemannian Geometry and Ricci flow in physics

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In the past thirty years or more, the mathematical fields of partial differential equations and Riemannian geometry have come together to create the field called geometric analysis. This was partly motivated by General Relativity, where from the very beginning this synthesis has yielded considerable progress. In recent years, there have been spectacular mathematical advances in certain areas of geometric analysis. An example is the Ricci flow, which is now understood to the point where it has yielded a claim of the hundred year old Poincare conjecture and the much broader Thurston conjecture, describing the nature of all smooth 3-dimensional compact objects ("manifolds"). The purpose of this project is to apply some of these advances to problems in physics. For example, the Ricci flow is also believed to describe, at least approximately, certain dynamical behaviour in string theory. A related flow promises to be important in the understanding of time-independent solutions of Einstein's general relativity theory. This project is aimed at exploring these connections.