Mathematical Sciences: Problems Related to Differential Geometry and Mathematical Physics

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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.