Mathematical Sciences: Low-Dimensional Geometry and Topology

小行星9704135 该项目是对低维几何和拓扑的多管齐下的研究，其中心长期目标是支持和建立几何化猜想，即每个3维流形都有一个正则分解成局部齐次黎曼度量。 第一个这些尖头是调查，部分由计算机辅助，几何Dehn填充空间的3流形。 几何Dehn填充插值流形之间的不同拓扑结构。 这个想法的一个适当的变化有可能连接所有可能的3-流形，并为它们配备几何分解。 第二个方面是研究绷紧叶理和基本层积的几何形状，以及它们与双曲结构和其他几何结构的关系。 这种方法有希望给一个共同的推广规范分解的表面同胚开发的瑟斯顿在1976年，和几何化的哈肯流形开发的瑟斯顿在70年代末和80年代初。 这个项目的其他分支涉及接触结构，confoliations和几何群论。 此外，该项目将探索低维几何和拓扑学与生物学，特别是遗传学的联系。 低维几何和拓扑是数学中一个美丽的领域，在过去的二十年里经历了巨大的发展和变革。 在学科内部发展的基础上，数学与科学其他领域的新联系和加强联系的外部发展则稍显滞后。 其中一些联系是直接的，比如我们宇宙形状的宇宙学问题，或者更实际的晶体形状问题。 其他的连接是间接的，不同的主题，如数据库，群论，遗传学，或计算化学。 任何可以用定量方法表述的主题都可以用几何方法（以及更流行的符号，代数和分析方法）进行研究，因此在任何特定情况下，低维几何和拓扑学的强大发现和工具都有合理的机会产生直接影响。 本项目的目标是进一步开发低维几何和拓扑工具，使其更易于移植，功能更强大，更具有通用性。 ***

9704135 Thurston This project is a multi-pronged investigation into low-dimensional geometry and topology with the central long-range goal of supporting and establishing the Geometrization Conjecture, namely, that every 3-dimensional manifold has a canonical decomposition into pieces with locally homogeneous Riemannian metrics. The first of these prongs is an investigation, partially aided by computer, of geometric Dehn filling spaces for 3-manifolds. Geometric Dehn fillings interpolate between manifolds with distinct topology. A suitable variation of this idea has the potential to connect all possible 3-manifolds and equip them with geometric decompositions. The second prong is an investigation of the geometry of taut foliations and essential laminations and their relationships to hyperbolic structures and other geometric structures. This approach has the promise to give a common generalization of the canonical decompositions of surface homeomorphisms developed by Thurston in 1976, and the geometrization of Haken manifolds developed by Thurston in the late 1970's and early 1980's. Additional prongs of this project involve contact structures, confoliations, and geometric group theory. In addition, the project will explore connections of low-dimensional geometry and topology to biology, particularly genetics. Low-dimensional geometry and topology is a beautiful area of mathematics that has undergone tremendous growth and transformation over the last two decades. Paralleling this internal development of the subject, lagging slightly behind, has been the external development of new connections and strengthened connections to other areas of mathematics and of science. Some of these connections are direct, like the cosmological issue of the shape of our universe or the more down-to-earth issue of the shapes of crystals. Other connections are indirect, to diverse topics such as databases, group theory, genetics, or computational chemistry. Any topic tha t can be formulated in a quantitative way can be studied with geometric methods (alongside the more prevalent symbolic, algebraic and analytic methods), so that in any particular case, the powerful discoveries and tools of low-dimensional geometry and topology have a reasonable chance to have a direct bearing. The aim of this project is further development of the tools of low-dimensional geometry and topology to make them more portable, more powerful and more universal. ***