Shoshichi Kobayashi will continiue his work on the geometry of holomorphic vector bundles. This is a topic of great interest in differential geometry, algebraic geometry and mathematical physics. The recent work on the classification of four dimensional manifolds has provided a marvellous example of the surprising interplay between different mathematical disciplines. Kobayashi's research is central to many of these exciting developments. A new aspect to his work is his development of a theory of infinite dimensional manifolds. This should prove to have applications to loop manifolds and therefore have implications in string theory. The first of Koabayashi's areas of research will be the moduli space of Einstein-Hermitian vector bundles. This involves the study of self-dual connections on these bundles. These are known to be minimizing for the Yang-Mills functional. He will attempt to study a combined moduli space comprising moduli of bundles with moduli of complex structures on the base manifold. The second area will continue his work on applications of hyperbolic complex analysis to complex analytic varieties with singularities. It is very likely that this will have applications in diophantine geometry. The final area of study will be the set of mappings from one manifold to another, this is itself an infinite dimensional manifold with a rich structure. This topic is particularly important at this time in view of the growth of interest in string theory.
