Low-Dimensional Geometry and Topology

AbstractAward：DMS-0103843项目负责人：罗峰项目负责人将重点研究Teichmuller理论和三维流形拓扑中的两个问题。在Teichmiller理论中，研究的目的是通过构造由黎曼曲面的平坦奇异度量一致化所产生的全纯函数来理解Teichmiller空间上的复杂结构。我们已经产生了许多自然定义的复值函数的Teichmuller空间。我们的目标是证明它们是全纯的。这将使我们更好地理解复杂的结构，这对Teichmuller理论至关重要。 在三维流形拓扑中，我们证明了任何非平凡的三维流形群对域F都有一个非平凡的SL（2，F）表示。我们已经将存在性问题转化为一个关于简单环如何在曲面中传播的问题。 随着我们对曲面知识的不断发展，人们可能最终会用曲面拓扑来解决这个问题。 SL（2，F）表示的存在性在三维流形拓扑中具有重要的意义，三维流形是一个空间，其中每个点都有一个类似于我们的真实的世界的小包围. 对所有三维流形进行分类是一个重要的数学问题。几何学是近几十年来三维流形理论发展的主要工具之一。特别是，曲面几何已被非常成功地用于理解三维空间。 本文的工作主要涉及三维流形的拓扑结构和曲面的几何结构。 我们试图用对称性理论（SL（2）表示理论）来理解三维流形的基本群，它是三维流形的一个重要不变量。 SL（2，C）表示理论近年来被许多拓扑学家成功地应用。我们的方法似乎是新的，在表面上使用简单的循环。 第二部分提出的工作地址的几何表面。曲面几何的主要问题之一是模空间问题。模空间问题要求，例如，是什么形状的空间的所有凸多面体看起来像一个立方体。 许多几何问题最好用模空间的拓扑和几何来表示。高亏格曲面的对应对象是Teichmuller空间。在对比的拓扑结构的Teichmuller空间这是很好地了解了约60年，几何的，它是少得多的理解。我们提出的工作是试图明确理解复杂的解析几何的Teichmuller空间。Teichmuller空间的复杂几何的明确描述不仅在数学上有应用，而且在物理学上也有应用，例如在弦理论中。

AbstractAward: DMS-0103843Principal Investigator: Feng LuoThe principal investigator will focus on two problems in theTeichmuller theory and 3-manifold topology. In Teichmullertheory, the aim of the investigation is to understand the complexstructure on the Teichmuller space by constructing holomorphicfunctions arising from flat singular metric uniformization of theRiemann surface. We have produced many naturally defined complexvalued functions on the Teichmuller space. The goal is to showthat they are holomorphic. This will give us a betterunderstanding of the complex structure which is of vitalimportance to the Teichmuller theory. In 3-manifold topology, wepropose to show that any non-trivial 3-manifold group has anon-trivial SL(2,F) representation for some field F. We havetranslated the existence problem into a problem concerning howsimple loops propagate in a surface. With the recent advance ofour knowledge on surfaces, one may eventually solve the problemusing surface topology. The existence of SL (2,F)representations will have many important consequences in3-manifold topology.A 3-manifold is a space in which every point has a smallsurrounding similar to our real world. It is an importantmathematical problem to classify all 3-manifolds. One of the maintool developed in recent decades in 3-manifolds theory is to usegeometry. In particular, the geometry of surfaces has been usedvery successfully in understanding the 3-dimensional spaces. Theproposed work addresses the topology of 3-manifolds and thegeometry of surfaces. We attempt to use the symmetry theory(SL(2) representation theory) to understand the fundamental groupof 3-manifolds which is a vital invariant of 3-manifolds. TheSL(2,C) representation theory has been used very successfully inrecent years by many topologists. Our approach seems to be newand uses simple loops on surfaces. The second part of theproposed work addresses the geometry of surfaces. One of the mainproblems on surface geometry is the moduli space problem. Themoduli space problem asks for, for instance, what is the shape ofthe space of all convex polyhedrons which look like a cube. Manygeometric problems are best expressed in terms of the topologyand geometry of the moduli space. The corresponding object forhigh genus surface is the Teichmuller space. In contrasts to thetopology of the Teichmuller space which is well understood forabout 60 years, the geometry of it is much less understood. Ourproposed work is an attempt to understand explicitly the complexanalytic geometry of the Teichmuller space. The explicitdescription of the complex geometry of the Teichmuller space willhave applications not only in mathematics but also in physics,for instance in string theory.