Mathematical Sciences: Twistor, Complex Manifolds, and      Differential Geometry
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Claude LeBrun will study a number of problems in or related                to Riemannian geometry in which complex-analytic techniques may                 be brought to bear by means of a twistor correspondence - that                  is, by thinking of the points of a Riemannian manifold as compact               complex submanifolds of some associated complex manifold. Several               of these problems, notably questions concerning solutions to                    Einstein's equations and the Yang-Mills equations, are motivated                by physics. Others are problems internal to geometry and analysis               concerning the existence or non-existence of solutions to                       particular partial differential equations of a geometric                        character.                                                                           There are three principal directions in which the research                 will proceed. First he will seek large categories of examples of                quaternionic manifolds. The second direction concerns further                   extensions of Witten's interpretation of Yang-Mills fields in                   terms of third order infinitesimal neighborhoods of embeddings.                 Such extensions have already been carried out by LeBrun in the                  case of self dual manifolds. The final part of the program                      involves defining a quaternionic analog of the notion of CR                     structure.
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{'Code': '1265', 'Text': 'GEOMETRIC ANALYSIS'}
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