Mathematical Sciences: Geometric Evolution Equations, Einstein Manifolds, Minimal Surfaces and Pseudo Holomorphic Curves

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Professor Ye will study evolution equations and applications to problems in Riemannian geometry. In particular he will study the Ricci flow, the Yamabe flow, and the harmonic mapping flow. The method of evolution equations has become an important method for finding critical points of geometric functionals and in understanding the dynamic structure of space. The evolution equations to be studied in this research can be understood physically by imagining a stretched rubber band immersed in a viscous fluid such as oil. The evolution equation for this example then describes the way in which the rubber band shrinks. The speed at which it shrinks at any point on the band is proportional to how much the band is curved at that point. Since so many physical problems involve deformation of curves or surfaces, the potential for application of this research is substantial.