This award will enable Prof. C. Herbert Clemens and colleagues of the University of Utah to collaborate with Prof. Shigefumi Mori of Nagoya University, Japan, over a period of two years. They will investigate certain aspects of higher-dimensional geometry, in particular the cohomological criteria for contractibility of curves on threefolds; the existence of rational curves on K-trivial threefolds; 3-dimensional flips; and extensions of results to dimensions greater than three. These problems, among others to be considered, all comprise algebraic, analytic, and topological aspects, and the proposed research brings together experts in all of these fields. The purpose of this research is to increase our understanding of geometry in three and more dimensions. To this end, the Japanese investigator and co-workers will pursue the questions raised using the thickening method and the flip conjecture, while the U.S. group will use the deformation theory of singularities, and Hodge theory. This work may be expected to have important implications in the field of mathematics, and also in the study of the deformation of the surfaces of materials.
