U.S.-Japan Cooperative Research: Joint Research in Higher- Dimensional Geometry

该奖项将使犹他大学的C. Herbert Clemens教授及其同事能够与日本名古屋大学的Shigefumi Mori教授进行为期两年的合作。他们将研究高维几何的某些方面，特别是关于三折曲线可收缩性的上同调准则；k平凡三叠上有理曲线的存在性；三维 翻转;并将结果扩展到大于3的维度。这些问题，以及其他需要考虑的问题，都包括代数、解析和拓扑方面，拟议的研究汇集了所有这些领域的专家。这项研究的目的是增加我们对三维及多维几何的理解。为此，日本研究人员和同事将使用增厚法和翻转猜想来解决提出的问题，而美国研究小组将使用奇点变形理论和霍奇理论。这项工作有望在数学领域以及材料表面变形的研究中产生重要的影响。

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.