Mathematical Sciences: Deformations of Complex Structures

数学科学：复杂结构的变形

• 批准号: 9003361
• 负责人: Irwin Kra
• 金额: $ 15.56万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-06-01 至 1993-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9003361&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Deformations; Complex; Structures

Two investigators will study Kleinian groups and Teichmuller spaces. One will analyze deformation spaces and cohomology of Kleinian groups, moduli of Riemann surfaces and their explicit representations, automorphic forms, and projective structures. The second will continue his investigation into the structure of Kleinian groups and hyperbolic 3-orbifolds emphasizing the classification of geometrically finite groups and combination theorems. The two investigators will work toward solving several problems related to the study of Kleinian groups. An example of such a group would be a collection of motions of the plane which preserve a periodic tiling. As elements of a group these must satisfy several axioms including one which requires the existence of an inverse element associated with each element of the group. If one element translates the plane by two units to the "right," its inverse element would translate the plane two units to the "left." The study of such motions has led mathematicians to the solutions of problems involving crystal structures.

两名研究人员将研究克莱尼群和泰希米勒空间。 我们将分析克莱因群的变形空间和上同调、黎曼曲面的模及其显式表示、自守形式和射影结构。 第二个将继续研究克莱因群和双曲三环折叠的结构，强调几何有限群的分类和组合定理。 两位研究人员将致力于解决与克莱因群研究相关的几个问题。 这种组的一个例子是保留周期性平铺的平面运动的集合。 作为群的元素，它们必须满足几个公理，其中包括一个要求存在与群的每个元素相关联的逆元素的公理。 如果一个元素将平面向“右”平移两个单位，则其逆元素会将平面向“左”平移两个单位。 对此类运动的研究使数学家能够解决涉及晶体结构的问题。