Mathematical Sciences: The Topology of Three-Manifolds

数学科学：三流形拓扑

• 批准号: 9204489
• 负责人: Charles Frohman
• 金额: $ 5万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-06-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204489&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topology; Three; Manifolds

Frohman intends to pursue the study of three-dimensional manifolds, especially topics associated with Heegaard splittings and minimal surfaces and the representation theory of knot groups and three-manifold groups. For instance, he is planning to study the finiteness of Heegaard splittings of bounded genus of three- manifolds up to homeomorphism. Another problem he is pursuing is the construction of functors that are similar to coupled topological quantum field theories using the intersection homology of moduli spaces of semistable holomorphic bundles with parabolic structure over a nonsingular algebraic curve. Three-dimensional manifolds are very natural objects for topologists to study, since (ignoring time) we live in a physical world of three dimensions. Also dimension three is the first dimension where many types of rich complexity arise. In fact, it is a notorious and paradoxical fact that some of the difficulties presented by three-dimensional topology disappear again in higher dimensions. For example, the celebrated Poincare conjecture about the characterization of the three-dimensional sphere remains as challenging as when it was first posed about the turn of the century, while the analogous propositions in higher dimensions have all now been settled. Frohman is exploring some of these mysteries and sharpening the available tools of the trade.

弗罗曼打算从事三维的研究， 流形，特别是与Heegaard分裂有关的主题 极小曲面与纽结群的表示理论 和三歧管组。 例如，他打算学习 有界亏格的Heegaard分裂的有限性 流形直到同胚。 他所追求的另一个问题是， 类似于耦合的函子的构造 拓扑量子场论 半稳定全纯丛模空间的同调 非奇异代数曲线上的抛物结构 三维流形是非常自然的物体， 拓扑学家研究，因为（忽略时间）我们生活在一个物理 三维世界。 三维空间也是第一个 在这个维度上，出现了许多类型的丰富复杂性。 实际上 是一个臭名昭著的矛盾的事实， 由三维拓扑结构呈现， 尺寸. 例如著名的庞加莱猜想 三维球体的特征 仍然像第一次提出转弯时一样具有挑战性 的世纪，而类似的命题在更高的 尺寸现在都已经确定了。 弗罗曼正在探索一些 这些奥秘和锐化可用的工具， 贸易