Research in Geometrical PDE

几何偏微分方程研究

• 批准号: 0103160
• 负责人: Jie Qing
• 金额: $ 5.4万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103160&HistoricalAwards=false
• 关键词: Research; Geometrical; PDE

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.

DMS - 0103160PI 摘要：杰庆提议的项目由两个相关的研究主题组成。第一部分是共形几何研究。第二部分是沿着几何流发展起来的奇点形成和结构研究的延续。对于共形几何部分，它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 4 维结构）并没有那么成功，因为缺乏正确的分析工具。 但似乎发现用四阶偏微分方程理论替代复分析对于研究共形几何是非常有前途的。因此，本研究的主要内容是利用这些新开发的分析工具进一步开发共形几何。拟议的研究采用基本且全面的方法来研究共形几何。它将显着增强和发展共形几何。 它还将提供一些方法来更好地理解 3 维和 4 维拓扑。对于这个拟议项目的第二部分，我们继续研究沿着表面谐波图的热流发展的奇点的形成和结构。了解跨越有限时间奇点的流的行为总是非常有趣的。更好地理解有限时间奇点被认为对于调和映射理论在拓扑学和物理学中的进一步应用非常有帮助。共形几何的发展有着非常悠久的历史和广泛的文献。它与现代物理学密切相关，正如我们在共形场论中看到的那样。特别是当量子引力和共形场论之间的对应关系在物理学中相对较好地理解时，它变得更加重要，需要数学基础，并刺激了共形几何的发展。因此，很明显，共形几何是现代科学的一个令人兴奋的前沿领域。科学中几乎所有事物的许多数学模型中都会自然地出现奇点。例如，当物理系统中的某些参数接近某些临界值时，就会出现奇点，例如超导和超流体中的金兹堡-朗道模型。研究演化系统中的奇点一直是偏微分方程理论的主要问题，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一点是，拟议的项目还产生了有利于加州大学圣迭戈分校数学系研究生项目的研究活动。