Abstract for DMS-97-03877 H. Alexander Alexander will study five problem areas in several complex variables: 1. Boundaries of Analytic Varieties and Linking. Given a smooth manifold M in complex space, when is M the boundary of a bounded analytic variety? Our aim is to formulate a global geometric condition that M bounds in terms of linking numbers. The main tool here is the Harvey and Lawson theory. 2. Analytic disks and Gromov's method. The method of Gromov's well-known 1985 paper on Lagrangian manifolds and pseudo holomorphic curves was adapted and applied in a paper of the proposer to study analytic disks with boundary in totally real manifolds. This study will be continued. 3. Hulls of sets in the torus. Well-known examples of Stolzenberg and Wermer give polynomial hulls without analytic structure. The proposer is interested in the problem of finding hulls, without analytic structure, of sets in the unit torus. 4. Fabre's Theorem. 5. Integrals involving analytic functions. Possible quantitative versions of Siu's recent lemma on the inheritance of non-integrability by restriction will be explored. This project involves the study of problems in complex analysis. The word "complex" here does not mean "complicated" but rather makes reference to "complex numbers." Complex numbers are nowadays routinely studied in high-school and a college course in "complex variables," because the material is so useful, is part of the basic undergraduate curriculum for engineers and scientists. The current project is directed toward better understanding "multidimensional complex space." Recent research in this area of mathematics continues to have applications; for example, in the engineering field of control theory.
