Global Riemannian Geometry

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The PI proposes to construct a Riemannian metric on the Gromoll-Meyer sphere with positive sectional curvature. This is a long standing open problem and is now especially relevant as Brendle-Schoen have shown that no exotic sphere admits a metric with positive complex sectional curvatures. As a separate specific project the PI also proposes to understand when quasi-Einstein and gradient soliton metrics are forced to be Einstein metrics. These questions go back to the earliest works on Einstein metrics, but have received new attention with Perel'man classification of certain gradient solitons in three dimensions. Understanding quasi-Einstein metrics is also important in general relativity as they occur as solutions to the Einstein field equations.In general terms the PI wishes to investigate what types of geometries are possible on specific topological objects. Two objects such as a doughnut and a tea cup are geometrically very different but topologically similar. The goal is to find the nicest possible geometries for specific topological objects. This is of interest to mathematicians, physicists, computer scientists,medical scientists and many other people as it is becoming increasingly clear that our flat Euclidean picture is not always the correct or even nicest model to use.