RUI: Complex Structures, Hyperbolic Invariants, Infinitesimal Currents and Intersection Numbers for Deformation Spaces

RUI：复杂结构、双曲不变量、无穷小电流和变形空间的交点数

• 批准号: 1102440
• 负责人: Dragomir Saric
• 金额: $ 15.4万
• 依托单位: CUNY Queens College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1102440&HistoricalAwards=false
• 关键词: RUI; Complex; Structures; Hyperbolic; Invariants

The Principal Investigator studies Teichmuller spaces of surfaces, both closed and open. The Teichmuller space of a surface is the space of all possible shapes of the surface, where a shape of the surface is the hyperbolic metric on a surface up to isometries homotopic to the identity. Therefore the invariants of hyperbolic metrics on a surface are used in the study of Teichmuller spaces. The PI?s approach is to first consider the Teichmuller space of the hyperbolic plane called the universal Teichmuller space as this space contains all other Teichmuller spaces. An invariant called a shear associated to an ideal triangulation of the hyperbolic plane is used to parameterize the universal Teichmuller space. The PI intends to continue his study of the universal Teichmuller space and related Teichmuller spaces in terms of these invariants called shears. In particular, the PI intends to describe the Weil-Petersson metric on the Teichmuller space of a finitely punctured closed surface in terms of shears on ideal triangulations of the surface, where triangulations can be both locally finite and locally infinite. He also intends to implement these formulas on the computer with the help of some undergraduate and masters students from Queens College. Another direction in applying shear invariants to the Teichmuller spaces is to find a parameterization of Takhtajan-Teo Teichmuller space in terms of shears and to find a formula for the Weil-Petersson metric in this space. This direction has possible applications to Sharon-Mumford?s approach to two-dimensional shape analysis in Computer Vision. The Quasifuchsian space of a closed surface supports a Weil-Petersson metric as well which is defined by taking the second partial derivative of the product of the Hausdorff dimension of the limit quasicircle and the Sullivan-Paterson measure. The PI intends to investigate the infinitesimal Sullivan-Paterson measures and their intersection numbers to obtain another expression for the Weil-Petersson metric on the Quasifuchsian space similar to the situation of the Fuchsian (Teichmuller) space.Riemann surfaces are two-dimensional objects which locally look like open subsets of a plane and that have transition maps which preserve angles. Each Riemann surface supports a unique hyperbolic metric in its class of conformal metrics. The PI studies the variations of hyperbolic metrics on a surface thought of as a single space of metrics called the Teichmuller space. The Teichmuller space is of interest in complex analysis, low-dimensional topology, dynamics, differential geometry and physics. One aspect of the project is related to the Computer Vision given by the approach of Sharon-Mumford as well as to the mathematical physics in the approach of Nag-Sullivan and Takhtajan-Teo. The project is building tools for study of the universal Teichmuller space and it has a potential for applications to the above mentioned fields. Another part of the project involves undergraduate and masters students from Queens College. The students participating in the project will be exposed to an active research agenda thus contributing to the human resource development in the sciences and engineering.

首席研究人员研究了闭合和开放的表面的Teichmuller空间。表面的Teichmuller空间是表面所有可能形状的空间，其中表面的形状是表面上的双曲线指标，直至同态均具有同一性。因此，在Teichmuller空间的研究中使用了表面上双曲线指标的不变性。 PI的方法是首先考虑称为通用Teichmuller空间的双曲线平面的Teichmuller空间，因为此空间包含所有其他Teichmuller空间。一种称为与理想三角剖分相关的称为剪切的不变性，用于参数化通用的Teichmuller空间。 PI打算继续研究这些不变的剪切物，并以这些不变的速度来研究。特别是，PI打算用有限刺穿的闭合表面的Teichmuller空间上的Weil-Petersson度量，该指标在表面理想的三角剖分上的剪切作用有限，其中三角剖分可以是局部有限的，也可以是本地无限的。他还打算在皇后学院的一些本科生和硕士学生的帮助下在计算机上实施这些公式。将剪切不变性应用于Teichmuller空间的另一个方向是，在此空间中找到了Takhtajan-Teo Teichmuller空间的参数化，并在此空间中找到一个公式。这个方向可能应用于沙龙·姆福德（Sharon-Mumford）在计算机视觉中进行二维形状分析的方法。封闭表面的准齐路空间也支持Weil-Petersson度量标准，该度量是通过将限量Quasicircle和Sullivan-Paterson量的Hausdorff尺寸的第二个部分衍生物和乘积定义的。 The PI intends to investigate the infinitesimal Sullivan-Paterson measures and their intersection numbers to obtain another expression for the Weil-Petersson metric on the Quasifuchsian space similar to the situation of the Fuchsian (Teichmuller) space.Riemann surfaces are two-dimensional objects which locally look like open subsets of a plane and that have transition maps which preserve angles.每个Riemann表面都支持其一类保形度量中的独特双曲线度量。 PI研究表面上的双曲线指标的变化是称为Teichmuller空间的单一指标空间。 Teichmuller空间在复杂分析，低维拓扑，动力学，差异几何形状和物理学中引起了人们的关注。该项目的一个方面与Sharon-Mumford的方法以及Nag-Sullivan和Takhtajan-Teo方法中的数学物理学有关的计算机视觉有关。该项目是为通用Teichmuller空间研究的工具，并有可能在上述领域应用。该项目的另一部分涉及本科生和皇后学院的硕士学生。参加该项目的学生将接触到积极的研究议程，从而为科学和工程的人力资源开发做出贡献。