Mathematical Sciences: Problems Related to Differential Geometry and Mathematical Physics

帕特里西奥·阿维尔斯将继续他在偏微分方程方面的工作，这些偏微分方程与微分几何和数学物理问题有关。这项工作涉及四个方面：完全黎曼流形中的Yamabe问题；液晶问题；两个流形之间的调和映射；半线性椭圆型方程正解的奇异性。紧凑的山部问题是由舍恩解决的。Aviles将尝试扩展他和mccowen的工作，他们在处理完全流形的情况下取得了一些进展。液晶的工作涉及到最小表面理论，并涉及到极限构型奇异集大小的研究。第三个主题集中于负曲率完全流形之间调和映射的Dirichlet问题。对半线性椭圆型方程正解的研究是由对杨-米尔斯方程的研究引起的。阿维莱斯将研究非孤立奇点。

Patricio Aviles will continue his work on partial differential equations connected with problems on differential geometry and mathematical physics. This work concerns four areas: the Yamabe problem in complete Riemannian manifolds; liquid crystal problems; harmonic maps between two manifolds; singularities of positive solutions of semi-linear elliptic equations. The compact Yamabe problem was solved by Schoen. Aviles will try to extend his work with McOwen where they made some progress in dealing with the case of complete manifolds. The liquid crystal work involves minimal surface theory and concerns an investigation of the size of singular sets of limiting configurations. The third topic focusses on the Dirichlet problem for harmonic maps between complete manifolds of negative curvature. Interest in positive solutions of semi-linear elliptic equations is motivated by study of the Yangs-Mills equations. Aviles is going to study non-isolated singularities.