Mathematical Sciences: Representations of Three-Manifold Groups

数学科学：三流形群的表示

• 批准号: 8911329
• 负责人: Andrew Casson
• 金额: $ 17.42万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1993-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8911329&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Representations; Three; Manifold

Andrew Casson will study aspects of the geometrization problem for 3-manifolds. He will attempt to improve the "Strong torus theorem" to conclude that irreducible 3-manifolds which contain essential singular tori either contain essential embedded surfaces of genus one or are Seifert fibered spaces. His method is related to one used by Tukia. He will also attempt to find good usable conditions on a 3-manifold guaranteeing the existence of representations of the fundamental group in PSL2(C). In particular, he will seek methods of finding hyperbolic or other geometrically interesting representations. Robion Kirby will continue studying Witten's program for defining an invariant of framed 3-manifolds via an "average" of the Chern-Simons invariants for SU(n) valued connections. He will also work on the diffeomorphism classification of smooth 4-manifolds obtained by "Gluck twists" on imbedded 2-spheres, or a generalization to immersed 2-spheres with one double point. Yakov Eliashberg will study invariants of Legendrian knots. All three projects concern different aspects of 3-dimensional manifolds, which are natural geometric objects describable locally by 3 independent coordinates. The structure of 3-manifolds has relevance to other parts of mathematics inasmuch as such manifolds occur as solution sets of ordinary or differential equations. Relevance to physical theories is also evident.

安德鲁卡森将研究几何化的各个方面， 3-流形的问题 他将努力改善 “强环面定理”得出不可约3-流形 包含本质奇异环面的环，或者包含本质 亏格为1或的嵌入曲面是塞弗特纤维空间。 他的方法与Tukia使用的方法有关。 他还将 试图在三维流形上找到好的可用条件 保证存在的表示， PSL 2（C）中的基本群。 特别是，他将寻求 求双曲线或其他几何形状的方法 有趣的表现。 Robion Kirby将继续研究维滕的计划， 定义一个不变量的框架3流形通过“平均”的 SU（n）值联络的Chern-Simons不变量 他 还将致力于光滑的微分同胚分类 4-流形获得的“格鲁克扭曲”嵌入2球， 或者推广到具有一个双点的浸入2-球面。 Yakov Eliashberg将研究Legendrian结的不变量。 所有三个项目都涉及以下不同方面： 3-维流形，这是自然的几何对象， 可以用3个独立的坐标来描述。 的 3-流形的结构与 数学，因为这样的流形作为解集出现 普通或微分方程。 与物理相关性 理论也是显而易见的。