Low-Dimensional Geometry and Topology

摘要奖：DMS-0103843主要研究人员：罗峰主要研究TeichMuller理论和3-流形拓扑中的两个问题。在TeichMuller理论中，研究的目的是通过构造由Riemann曲面的平坦奇异度量均匀化产生的全纯函数来了解TeichMuller空间上的复结构。我们在TeichMuller空间上产生了许多自然定义的复值函数。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结构，这对泰希穆勒理论至关重要。在3-流形拓扑中，我们证明了任一非平凡的3-流形群对域F都有一个非平凡的SL(2，F)表示。我们将存在问题转化为一个关于环在曲面上如何传播的问题。随着我们最近对曲面知识的发展，人们最终可能会用曲面拓扑学来解决这个问题。SL(2，F)表示的存在将在3-流形拓扑中产生许多重要的结果。3-流形是一个空间，其中每个点都有一个类似于我们现实世界的小环境。3-流形的分类是一个重要的数学问题。近几十年来发展起来的三维流形理论的主要工具之一是使用几何方法。特别是，曲面的几何已经被非常成功地用于理解3维空间。所提出的工作涉及3-流形的拓扑和曲面的几何。我们试图用对称理论(SL(2)表示理论)来理解3-流形的基本群，它是3-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非常成功地应用。我们的方法似乎是新的，在曲面上使用简单的循环。拟议工作的第二部分解决了曲面的几何问题。曲面几何中的一个主要问题是模空间问题。例如，模多面体空间问题是问，所有看起来像立方体的凸多面体的空间形状是什么。许多几何问题最好地用模空间的拓扑和几何来表示。对应于高亏格曲面的对象是TeichMuller空间。泰希穆勒空间的拓扑学已被人们熟知了约60年，与之相比，人们对它的几何学却知之甚少。我们提出的工作是试图明确地理解TeichMuller空间的复解析几何。Teichmuller空间的复几何的显式刻画不仅在数学上有应用，而且在物理上也有应用，例如在弦论中。

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.