Proposal: DMS 9803399 Principal Investigator: Jie Qing
The proposed project consists of two related research topics. The first is focused in the investigations in developing and using nonlinear analysis of partial differential equations to study formations of singularities in various problems, arising from theoretical physics mostly in differential geometrical settings, including harmonic maps, Yang-Mills-Higgs theory, harmonic map flows, and curvature driven flows. The main approach is based on scaling methods. One problem that I am working on now is to understand some finer structure of the compactified spaces of solutions (harmonic maps). This is an essential problem for one to understand the formation and nature of singularities. The second part is the investigation of the interplay of geometry and analysis. The central problems are to understand the best constants in sharp Sobolev type inequalities in terms of the isoperimetric constants, and to understand how the local analytic quantities are related to the global geometry even the topology. One of the approach is to study the spectral theory of various natural differential operators associated with the geometry, especially the conformally covariant geometric differential operators.
The formation and nature of singularities in nature are of enormous interests to human beings. The kind of singularities that will be studied in the proposed project are particularly of great interests to the material sciences and meteorology. For instance, the formation and nature of singularities in Ginzburg-Landau problems as a particular case of the Yang-Mills-Higgs theory was proposed to study super-conductivity and other super-fluids phenomena. The effect of the scale that one uses to describe the system turns out to be very important. This explains that to understand anything that survives under changes of scales is very crucial. Therefore our main approach is based on the scaling methods. The second research topics in the proposed project is basically aiming at developing and understanding some mathematical tools that are expected to be essential for the first topics. The proposed project is also closely tied with the graduate program in the department of Mathematics at UCSC by a plan to generate research activities to the benefit of graduate student interested in this area.
