Low-dimensional geometry and topology

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其他数学和科学领域的相互联系。他将继续与Allen Hatcher合作，分析支链聚合物空间的拓扑结构，推广多于或等于8个等大小原子的构型，第五同伦群具有高秩的结果。Thurston将扩展与Hass和Thompson的合作，研究Heegaard分裂的整体几何、桥结点数、组发电机组，以及连续泛化的测地线流桥测量。Thurston还将研究近似有限k生成群的空间几何：群的Cayley图是有限群的Cayley图的极限。近似有限k生成群的空间是紧致的，重要的闭子集是可数的。这个空间给出了有限生成群的有限商和剩余有限的对偶性质。数学家对二维和三维空间现象的理解经历了一场戏剧性的革命，在佩雷尔曼解决瑟斯顿的几何化猜想（包括著名的庞加莱猜想）时达到高潮，为40年前最疯狂的梦想远远超出的问题提供了美丽的几何答案。在这一革命性变化中发展起来的几何见解和工具主要在一个专门的社区内被理解，但瑟斯顿感兴趣的是将它们的解释力从三维拓扑学扩展到其他领域的强大潜力，包括数学内部和外部。一项倡议是分析支链聚合物的空间，这是一种理想化的理论，展示了一些有趣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。这个项目的另一个创举涉及到所有可能有限群空间的几何。有限群在数学和科学中无处不在，它们描述了塑造我们世界的可见和隐藏的对称性。主要的（和强大的）方法是通过代数和表示理论的工具来研究有限群。我们将研究有限群的几何，以阐明从代数的角度来看不容易看到的现象。

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.