Mathematical Sciences: Complex Geometry and Mathematical Physics

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The two investigators will study a variety of geometric structures which have their origins in mathematical physics as well as complex geometry. These include projective algebraic manifolds, complex manifolds, quaternionic Kahler manifolds, and supermanifolds. The investigation will include a study of the momentum mapping in connection with quaternionic Kahler reduction, self-dual manifolds, and the twistor representation of quaternionic Kahler structures. The dynamics of the static self- dual solutions of the two-dimensional Yang-Mills-Higgs equations will be studied with the use of a computer. The two investigators will continue their studies of the new "supergeometry" which was created as part of gauge theory to understand the relationship between cosmology and subatomic physics. One will continue his work on the twistor geometries in Nigel Hitchin's framework. The second will use "quaternionic quotients" to derive further knowledge of complete noncompact manifolds with unusual spin holonomy. This last is a property of foliations and measures their shearing tendencies.