Mathematical Sciences: Problems in Potential Theory

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9400687 Wu This award supports mathematical research on problems arising from the classical theory of complex analysis and potential theory approached from several new points of view. The work concerns investigations into the nature of null sets for doubling measures and related problems on quasiconformal mapping. In particular the question of whether null sets of quasisymmetric homeomophisms on the unit circle are invariant under bilipschitz homeomorphisms. Work will also continue on harmonic measures for uniformly elliptic partial differential operators. Particular emphasis will be placed on building operators on the positive orthant by genuinely n-dimensional methods so that the support of the corresponding harmonic measures have small Hausdorff dimensions. Effort will be made to investigate practical ways of estimating expected lifetime for conditional Brownian motion for domains satisfying a capacity density condition. A new characterization of such domains is expected to result. Research in complex analysis and potential theory focuses on problems related to real and complex functions of several variables which satisfy certain partial differential equations. The theory is highly geometric and methods for attacking the many important problems are drawn from a multitude of mathematical theories such as probability, differential geometry, differential equations and functional analysis. ***