Topics in Quasiconformal mappings and in PDE

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The problems studied in this research project have as common theme the qualitative study of critical points of functionals arising from geometry and of their gradient flows. More specifically the principal investigator will continue to study weak solutions to a family of nonlinear partial differential equations (PDE) that are the sub-Riemannian analogues of the mean curvature flow PDE, the minimal surface PDE, and the conformal n-Laplacian. The principal investigator will also study a geometric function theory/nonlinear PDE approach to the boundary regularity of biholomorphic mappings between strongly pseudoconvex sets and pursue the study of extremal quasiconformal mappings in space. Both in the natural sciences and in engineering it is important to study equilibrium states of complex systems. Mathematics allows to model such states as critical points of energy functionals and to study their properties by analyzing certain differential equations associated to these functionals. The properties of the solutions to these equations depend on a 'background geometry' that models such real-life features as the non-homogeneity of materials, or the presence of constraints (such as in the motion of robot arms, etc. etc. ...). One of the most ubiquitous instances of such 'background geometry' is sub-Riemannian geometry, modeling spaces where motion is possible only along a given set of directions, as in applications to complex analysis, motion of robot arms and control theory, quantum computing, satellites, quantum mechanics and in the structural functions of the mammalian visual cortex. The PI will pursue this work together with his students as well as his collaborators. The study of sub-Riemannian geometry has been driven since its origins by "real world" problems and the PI lectures to a wide variety of audiences from the post-graduate level to the high school level.