Mathematical Sciences: Differential Geometry

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The Principal Investigator will study a number of questions concerning maps of surfaces into a compact manifold. The maps under consideration will satisfy an elliptic partial differential equation. In particular he plans to study the structure of the moduli space of harmonic maps of the two-sphere into a homogeneous Kaehler manifold. He will also investigate minimal surfaces in the complex projective plane and study pseudo holomorphic curves with a view to applications in symplectic and other geometries. Jon Wolfson has established himself as one of the world leaders in the theory of harmonic maps. He has accumulated a vast repertory of techniques, such as the structure of semi-simple Lie groups, complex algebraic geometry and partial differential equations. The subject of harmonic maps has recently acquired prominence following the discovery of interesting global structure theorems on their moduli spaces. In addition the connections with minimal surface theory in low dimensional topology and with sigma-models in mathematical physics have led to a spectacular growth in the number of researchers in this area.