Geometric flows and four-dimensional geometry

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This project lies at the interface of geometry and partial differential equations. The PI will further the study of natural geometric evolution equations designed to understand aspects of four dimensional geometry and topology. One project is to understand the singularity formation and long time behavior of geometric flows on almost Hermitian manifolds, previously introduced by the PI in joint work with G. Tian. One aspect of this is to fully understand how the long time behavior of the pluriclosed flow relates to the classification of complex surfaces. Another project is to understand the subcritical behavior of the gradient flow of the Yang-Mills energy of a Riemannian metric. One major goal here is to understand the possible singularities which can develop on four-manifolds with small initial energy.The use of geometry and curvature in understanding spaces is a fundamental idea in mathematics and science. Frequently the relevant curvature properties are closely related to equations relating to the laws of nature. Geometric evolution equations of the kind proposed here arise frequently in various physical contexts, and see application to various areas including image analysis and material science. The proposed research will advance fundamental geometric and analytic ideas which are the foundation of such applications.