Mathematical Sciences: Low-Dimensional Geometry and Topology

这个项目是对低维几何和拓扑的多管齐下的研究，其中心目标是支持和建立几何化猜想，即每个三维流形都有一个正则分解成局部齐次黎曼度量的碎片。第一个分支是对3流形的几何Dehn填充空间的研究，部分由计算机辅助。几何Dehn填充在具有不同拓扑结构的流形之间进行插值。这个想法的一个合适的变体有可能将所有可能的3-流形连接起来，并为它们配备几何分解。第二个分支是紧叶理和基本层合的几何研究，以及它们与双曲结构和其他几何结构的关系。这种方法有希望给出Thurston在1976年提出的曲面同胚的正则分解，以及Thurston在1970年代末和1980年代初提出的Haken流形的几何化的一般推广。这个项目的其他方面包括接触结构、组合和几何群论。此外，该项目将探索低维几何和拓扑学与生物学的联系，特别是遗传学。低维几何和拓扑学是数学中一个美丽的领域，在过去二十年中经历了巨大的增长和转变。与这门学科的内部发展并行，稍微落后的是与数学和科学其他领域的新联系和加强联系的外部发展。其中一些联系是直接的，比如宇宙形状的宇宙学问题，或者更实际的晶体形状的问题。其他连接是间接的，涉及不同的主题，如数据库、群论、遗传学或计算化学。任何可以用定量方式表述的主题都可以用几何方法（以及更流行的符号、代数和解析方法）来研究，因此，在任何特定情况下，低维几何和拓扑的强大发现和工具都有合理的机会直接影响。该项目的目标是进一步发展低维几何和拓扑工具，使其更便携、更强大、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. ***