Geometry, Measures, and Integral Operators for Boundaries of Complex Domains

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This project will focus on a variety of topics in complex analysis and related geometric theories, including the following interconnected subjects: i) the study of invariant geometric properties of real hypersurfaces in complex projective space; ii) the study of the Leray transform, a multivariate generalization of the Cauchy transform, with special attention to the information it provides about pairing of Hardy spaces on dual hypersurfaces in complex projective space; iii) continuation of a project with Mike Bolt to investigate geometric and analytic topics relating to a variety of invariant arc lengths including that of Laguerre; iv) extension and application of the theory of Fefferman's equi-holomorphically invariant boundary measure with particular attention to an associated variational problem. Functions of complex variables arise naturally and play an important role in many parts of mathematics, physics and engineering. This project is focused on issues of boundary geometry and analysis which are fundamental in nature. In particular, the study of the Leray transform and related matters has substantial connections with the Laplace transform, an important tool in the study of partial differential equations. The project borrows from (and, it is hoped, will contribute to) a variety of geometric theories of independent interest. Finally, the project will involve work with younger investigators and thus will help to ensure that expertise in this area is available in the future.