Abstract Proposal: DMS-9803192 Principal Investigator: Simon Donaldson The main topic of this project involves the application of methods from complex geometry to symplectic topology. The investigator will develop a general procedure for translating problems in this area into combinatorial questions involving the monodromy of a family of codimension 2 submanifolds. It is expected that this will apply, in principle, both to the classification problem for symplectic manifolds and also to Lagrangian submanifolds and symplectomorphisms. Once the general foundations are in place applications will be considered: the question here will be to see if the combinatorial problems can be cast into a tractable form. Particular attention will be paid to the role of the Floer homology groups obtained from the monodromy. One subsidiary topic in the project involves research into the geometry of Kahler metrics: the specific goals here are to prove the existence of certain geodesics in the space of Kahler metrics, and apply these to Calabi's extremal metric program. This existence question is a version of the Dirichlet problem for the homogeneous Monge-Ampere equation, a topic of independent interest. The other subsidiary topic involves research into manifolds with exceptional holonomy groups, and particularly the search for new examples, obtained using complex 3-folds as building blocks. Complex numbers, made up of real and imaginary components, are fundamental throughout mathematics. In geometry, it has been realised since the middle of the last century that properties of the complex number system are intimately bound up with the geometry and topology of 2-dimensional spaces. The elaboration of this theme, and its extension to higher dimensions, has been one of the main achievements of twentieth century mathematics. The ideas have many contacts with numerous branches of Mathematical Physics, including the theory of potentials and fields, quantum theory and relativity. Mat hematically, many of the questions come down to the detailed analysis of nonlinear partial differential equations. The proposed research will contribute to this brood development, focusing on a number of specific and topical questions, in all of which complex variables play a key role.
