Lipman Bers, one of the founding fathers of Teichmuller theory, will continue his work on the strong deformation spaces of Riemann surfaces with nodes. He will attempt to extend the theory of Riemann surfaces with nodes to surfaces of infinite Poincare area. This is connected with the study of infinite dimensional deformation spaces and moduli spaces which are currently of interest to string theorists. He will also continue his investigations of the complex boundaries of Teichmuller spaces. Masatake Kuranishi's research will continue his work on the sectional curvature of a certain metric on the cartesian product of the two-sphere with itself. The objective is to deform this metric in such way as to obtain one of positive sectional curvature. If successful this will provide a positive answer to a conjecture of Hopf. Kuranishi's second topic of investigation is to bring the geometry of CR manifolds into the calculus of Fourier integral operators with complex phase on strongly pseudoconvex manifolds. Duong Phong will carry out research in two areas. These concern oscillatory integrals in Fourier analysis and the connections between string theory and Riemann surfaces. The former comprises an investigation of the role of geometry in the study of singular kernels. The latter will be a continuation of earlier work in which Phong, together with collaborators, has identified the scattering amplitudes of strings with the Weil- Peterson measure on the moduli space. Hubert Goldschmidt will work on problems in Riemannian geometry related to integral geometry and the deformation theory of Riemannian metrics. In particular his research will be concerned with rigidity questions arising from the Blaschke problem for compact symmetric spaces. This will be studied in the context of the Radon transform for tensors on these spaces.
