Mathematical Sciences: Several Complex Variables

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David Barrett will work on various topics relating to function theoretic and geometric properties of smooth bounded domains lying in n-dimensional complex space and in more general manifolds. Several of the research topics concern questions which are fairly well understood on domains which satisfy certain positivity conditions on the Levi form but are poorly understood on more general domains. The questions raised about approximation theory in standard function spaces, regularity of the Bergman projection, and singularities of biholomorphic and proper maps fall into this category. Barrett will also investigate problems relating to domains and CR manifolds satisfying very strong conditions on the Levi form. These include questions about global embedding of strictly pseudoconvex three dimensional CR manifolds and about global properties of Levi-flat hypersurfaces.