Research in Geometrical PDE

DMS -0103160 PI摘要：Jie Qing拟议的项目包括两个相关的研究课题。第一部分是共形几何的研究。第二部分是对沿着几何流发展的奇点的形成和结构研究的继续。共形几何部分以曲面理论作为高维几何的指导思想。在我们看来，共形平面结构理论，例如四维空间的共形平面结构理论，由于缺乏合适的分析工具，一直没有取得很大的成功。 但是，用四阶偏微分方程理论代替复分析似乎是研究共形几何的一个很有前途的方法。因此，本研究部分的主要工作是利用这些新发展的解析工具进一步发展共形几何。本文的研究是对共形几何研究的一个基本而全面的方法。它将显著地增强和发展共形几何。 它还将提供一些方法来更好地理解三维和四维拓扑结构。对于这个项目的第二部分，我们继续我们的研究，在研究的形成和结构的奇性发展沿着热流调和映射从表面。理解有限时间奇点上的流动行为总是很有趣的。对有限时间奇异性的深入理解有助于调和映射理论在拓扑学和物理学中的进一步应用。共形几何的发展有着悠久的历史和广泛的文献。正如我们在共形场论中所看到的那样，它与现代物理学有着密切的联系，特别是当量子引力与在物理学中相对较好理解的共形场论之间的对应关系需要数学基础时，它变得更加重要，同时也促进了共形几何的发展。因此，很明显，保形几何是一个令人兴奋的前沿现代科学。奇点在科学中几乎所有事物的许多数学模型中自然发展。例如，当物理系统中的某些参数接近某个临界值时，奇异性就会产生，如超导和超流体中的Ginzburg-Landau模型。发展系统的奇异性研究一直是偏微分方程理论中的重要问题，在许多物理和工程领域有着广泛的应用。这方面的任何重要分析进展都将极大地吸引所有相关领域专家的关注。最后，但并非最不重要的是，拟议的项目也产生了研究活动，以利于UCSC数学系的研究生课程。

Abstract for DMS - 0103160PI: Jie QingThe proposed project consists of two related research topics. The first part is research in conformal geometry. The second part is a continuation of the research of formation and structure of singularity developed along geometric flows. For conformal geometry part it takes surface theory as the guideline of geometry in higher dimension. The theory of conformally flat structure, in 4 dimension for instance, has not been so successful because of the lack of right analytic tools, in our view point. But it seems that some replacement of complex analysis by theory of fourth order PDE is found to be very promising to study conformal geometry. Therefore the main thread in the part of this proposed research is to further develop conformal geometry with those newly developed analytic tools. The proposed research takes a fundamental and comprehensive approach to the study of conformal geometry. It will significantly enhance and develop the conformal geometry. It will also provide some ways to better understand topology in 3 and 4 dimension. For the second part of this proposed project we continue our research in the study of formation and structure of singularity developed along the heat flow for harmonic maps from surfaces. It is always very interesting to understand the behavior of the flow across the finite time singularity. The better understanding of the finite time singularity is believed to be very helpful in further applications of the theory of harmonic maps in topology and physics.The development of conformal geometry has a very long history and extensive literature. It is intimately tied withmodern physics as we have seen it in conformal field theory.Particularly it becomes even more important as the correspondence between quantum gravity and conformal field theory relativelywell understood in physics demands mathematical foundation, aswell as stimulates development in conformal geometry. Thereforeit is clear that conformal geometry is an exciting frontier ofmodern sciences. Singularities naturally develop in many mathematical models for almost everything in sciences. Singularity,for instance, develop when some parameters in the physical system approach certain critical values, like Ginzburg-Landau model in super-conductivity and super-fluids. To study the singularities in evolution systems have been the major problems in the theory of partial differential equations with tremendous applications to many physical and engineering fields. Any essential analytic progress in this line will greatly attract attention from experts in all related areas. At last, but not the least, the proposed project is also generating research activities to the benefitof the graduate program in the Department of Mathematics at UCSC.