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

这一奖项将使犹他大学的C.Herbert Clemens教授及其同事能够在两年内与日本名古屋大学的Shigefumi Mori教授合作。他们将研究高维几何的某些方面，特别是三重曲线可收缩的上同调准则；K-平凡三重曲线上有理曲线的存在；三维反转；以及结果到三维以上的推广。除了其他需要考虑的问题外，这些问题都包括代数、分析和拓扑方面，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在三维和更多维度上对几何的理解。为此，日本的研究人员和同事将使用增厚方法和翻转猜想来追问所提出的问题，而美国的研究小组将使用奇点形变理论和霍奇理论。这项工作可望在数学领域以及材料表面变形的研究中具有重要意义。

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.