Mathematical Sciences: Low-Dimensional Geometry and Topology

9704135瑟斯顿这个项目是对低维几何和拓扑的多管齐下的研究，中心的长期目标是支持和建立几何化猜想，即每个三维流形都有一个具有局部齐次黎曼度量的正则分解成碎片。第一个部分是在计算机的辅助下，研究三维流形的几何Dehn填充空间。几何Dehn填充在具有不同拓扑的流形之间进行内插。这一思想的适当变化有可能将所有可能的三维流形连接起来，并用几何分解来装备它们。第二个方面是对紧凑的叶理和基本纹层的几何以及它们与双曲结构和其他几何结构的关系的研究。这个方法有希望给出一个共同的推广，由瑟斯顿在1976年发展的曲面同胚的标准分解，以及由瑟斯顿在1970年代末的S和1980年初的S发展的哈肯流形的几何化。这个项目的其他方面涉及接触结构，褶合和几何群论。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。低维几何和拓扑学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增长和变化。与这一学科的内部发展相平行的是，与数学和科学的其他领域的新联系和加强联系的外部发展略有滞后。其中一些联系是直接的，比如我们宇宙的形状的宇宙学问题，或者更实际的晶体形状问题。其他的联系是间接的，与不同的主题，如数据库，群论，遗传学，或计算化学。任何可以定量表述的主题都可以用几何方法来研究(与更流行的符号、代数和分析方法并驾齐驱)，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。本项目的目的是进一步开发低维几何和拓扑工具，使其更可移植、更强大和更通用。***

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. ***