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

叶教授将研究演化方程及其在黎曼几何问题中的应用。他将特别研究里奇流、山部流和谐波映射流。演化方程方法已成为寻找几何泛函临界点和理解空间动态结构的重要方法。本研究中要研究的演化方程可以从物理上理解，想象一个被拉伸的橡皮筋浸入像油这样的粘性流体中。这个例子的演化方程描述了橡皮筋收缩的方式。它在带上任何一点收缩的速度与带在该点弯曲的程度成正比。由于许多物理问题涉及曲线或曲面的变形，因此本研究的应用潜力是巨大的。

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.