Low-dimensional geometry and topology

低维几何和拓扑

• 批准号: 0910516
• 负责人: John Hubbard
• 金额: $ 57.49万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0910516&HistoricalAwards=false
• 关键词: Low; dimensional; geometry; topology

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Thurston will investigate several areas of low-dimensional geometry and topology and their interconnections with other areas of mathematics and science. He will continue a collaboration with Allen Hatcher to analyze the topology of the space of branched polymers, extending the result that for configurations of more than or equal to 8 equal-size atoms the fifth homotopy group has high rank. Thurston will extend joint work with Hass and Thompson to investigate the global geometry of Heegaard splittings, bridge number of knots, generating sets for groups, as well as a continuous generalization, bridge measure for geodesic flows. Thurston will also investigate the geometry of the space of approximately finite k-generated groups: groups whose Cayley graphs are limits of Cayley graphs of finite groups. The space of approximately finite k-generated groups is compact, and important closed subsets are countable. This space gives insight into finite quotients of finitely-generated groups, and the dual property of residual finiteness.Mathematician's understanding of 2-dimensional and 3-dimensional spatial phenomena has undergone a dramatic revolution culminating in Perelman's solution to Thurston's geometrization conjecture (which includes the famous Poincare conjecture), giving beautiful geometric answers to questions far beyond the wildest dreams of 40 years ago. These geometric insights and tools developed during this revolutionary change are understood mainly within a specialized community, but Thurston is interested in the strong potential for extending their explanatory power beyond three-dimensional topology into other domains, both inside and outside mathematics proper. One initiative is to analyze the space of branched polymers, an idealized theory that exhibits some interesting and unexpected topological phenomena. We hope the idealized topological theory will ultimately contribute to understanding real molecules. Another initiatives in this project involve the geometry of the space of all possible finite groups. Finite groups are pervasive throughout mathematics and science as descriptors of the symmetries both visible and hidden that shape our world. The predominant (and powerful) approach to finite groups is through tools of algebra and representation theory. We will investigate the geometry of finite groups to elucidate phenomena that are not readily seen from the algebraic point of view.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。瑟斯顿将研究低维几何和拓扑学的几个领域，以及它们与数学和科学的其他领域的相互联系。他将继续与艾伦哈彻合作，分析分支聚合物空间的拓扑结构，扩展结果，即对于大于或等于8个等大小原子的配置，第五同伦群具有高秩。 瑟斯顿将扩大联合工作与哈斯和汤普森调查全球几何的Heegaard分裂，桥梁数量的结，发电机组的群体，以及一个连续的推广，桥梁措施测地线流量。 瑟斯顿还将调查几何空间的近似有限k-生成群：集团的凯莱图的限制凯莱图的有限群。近似有限k-生成群的空间是紧的，重要的闭子集是可数的。数学家们对二维和三维空间现象的理解经历了一场戏剧性的革命，最终以Perelman解决Thurston的几何化猜想而告终（其中包括著名的庞加莱猜想），给出了美丽的几何答案的问题远远超出了40年前最疯狂的梦想。这些几何的见解和工具，在这一革命性的变化，主要是在一个专门的社区内理解，但瑟斯顿感兴趣的是强大的潜力，将其解释力扩展到其他领域以外的三维拓扑结构，无论是内部和外部的数学正确。 其中一项倡议是分析支化聚合物的空间，这是一种理想化的理论，表现出一些有趣的和意想不到的拓扑现象。我们希望理想化的拓扑理论将最终有助于理解真实的分子。该项目的另一个举措涉及所有可能的有限群的空间几何。 有限群在数学和科学中无处不在，它们是塑造我们世界的可见和隐藏的对称性的描述符。研究有限群的主要（也是强大的）方法是通过代数和表示论的工具。我们将研究有限群的几何，以阐明从代数观点不容易看到的现象。