Some impact of topology on variational problems

PI: S.-Y。摘要：拓扑对变分问题的影响在本课题中，我们将重点研究涉及向量值映射的变分问题。特别地，我们将研究具有适当拓扑信息的映射的能量泛函的上界和下界，光滑映射的Sobolev映射的弱序列逼近以及一些变分问题解的奇异集结构。从几何测量理论和微分代数拓扑的各种工具被用于处理这些问题。另一方面，我们处理这些问题的方法揭示了这些主题之间的新关系。涉及向量值映射的变分问题在分析、几何和物理中自然出现。除了它们的数学兴趣和重要性外，它们对数学物理中的各种模型也是至关重要的。典型的例子包括超导、磁性材料和液晶模型。这一领域的研究将有助于更好地理解某些类型的物理现象，并为数学在这一方向上的发展提供新的动力。

PI: S.-Y. Alice Chang and Fengbo Hang, Princeton UniversityDMS-0209504Abstract:Some impact of topology on variational problemsIn this project, we will focus on variational problems involving vector- valued maps. In particular, we will study upper and lower bounds ofthe energy functional for maps with suitable topological information,the weak sequential approximation of Sobolev maps by smooth maps andthe structure of singular sets for solutions of some variationalproblems. Various tools from geometric measure theory and differentialand algebraic topology are used in handling these issues. On the otherhand, our approach to the problems sheds light on new relations amongthese topics.Variational problems involving vector-valued maps appear naturally in analysis, geometry and physics. Besides their mathematical interests and importance, they are crucial to various models in mathematical physics. Typical examples include models for superconductivity, magnetic material and liquid crystals. The research in this area will lead to better understanding of certain type of physics phenomenon as well as new impetus in mathematics in this direction.