Geometric PDEs, Curvature, and Symmetry

Abstract: Curvature is an invariant of a Riemannian manifold that measures the local defect from being Euclidean.  One manifestation of curvature is the Ricci tensor, which is a second order differential operator on the Riemannian metric, and heuristically a Laplacian of the metric.  The Ricci tensor features in several geometric PDEs that are both of intrinsic interest and have proven extremely successful in solving problems in geometry and topology.  Symmetry assumptions can often make these equations more tractable.  On the other hand, certain classes of solutions display a tendency to be more symmetric than initially assumed.  In my talk I will introduce the geometric concepts involved and present some of my results that illustrate the role of symmetry.

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