Research in Geometrical PDE

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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.