Mathematical Sciences: Research in Circle Packings, Quasiconformal Geometry and Low Dimensional Geometric Topology

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The principal investigator will study conformal moduli and the energy of incompressible flows, and the relationship between circle packings and the geometry of Riemann surfaces. The first project involves an investigation into how knot types of field lines restrict arbitrary decreases in the energy. The second project will be a continuation of earlier work on the convergence of derivatives for hexagonal packing schemes. The principal investigator will carry out two projects in the area of geometric complex analysis. The first involves a study of the behavior of fluid flow. In particular, he will analyze the manner in which a flow loses energy or slows down when flow lines are knotted. The second project involves the relationship between how a circle can be packed with smaller circles, and aspects of conformal complex analysis.