Mathematical Sciences: Polynomially Convex Hulls and Evolution of Pseudoconvex Sets by Levi Curvature

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ABSTRACT Slodkowski Slodkowski will continue working on polynomial hulls improving on understanding the structure of hulls of families of Jordan arcs under low regularity assumptions. The first part of the proposal concentrates on a series of problems generated by the crucial question when is such a hull a topological hypersurface. Slodkowski plans to investigate them under regularity assumptions formulated in terms of quasiconformal geometry. Realization of this program will yield applications to holomorphic motions, among others. The second part of the proposal concerns the evolution of subsets in higher-dimensional space, defined in terms of weak solutions of certain degenerate nonlinear PDE's involving the Levi curvature. The main focus is on analyzing the evolution of pseudoconvex hypersufaces and of pseudoconcave sets, in order to understand the structure of weakly pseudoconvex domains and of polynomial hulls. A successful completion of this proposal would solve, among other things, important problems on holomorphic motions, which would have significant applications to the theory of dynamical systems and to Teichmuller theory. The theory of the dynamical systems, which is developing very rapidly at present, has important applications to to many applied problems. The Teichmuller theory has not only fundamental implications for geometry, but it is also an important tool for string theory, one of the newest theories of the elementary particles' physics. Zbigniew Slodkowski