Geometric variational problems and nonlinear partial differential systems

我们将研究非线性偏微分系统、桥接分析、微分几何和数学物理中的几个重要问题。调和映射是极小化Dirichlet能量的映射的原型。液晶构型将带值调和映射推广到二维球面。杨-米尔斯方程起源于粒子物理学。我们将在以下几个方面做出基础性的贡献：弱调和映射的正则性问题和能量极小性，液晶平衡系统的弱解，Yang-Mills热流和奇异Yang-Mills联络。

We will investigate several important problems on non-linear partial differential systems, bridging analysis, differential geometry and mathematical physics. Harmonic maps are the prototype of maps minimizing the Dirichlet energy. The liquid crystal configuration generalizes the harmonic map with values into two dimensional spheres. The Yang-Mills equations originated from particle physics. We will make fundamental contributions to these topics: Regularity problem and energy minimality of weakly harmonic maps, Weak solutions of the liquid crystal equilibrium system, Yang-Mills heat flow and singular Yang-Mills connections.