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

首席研究员研究表面的Teichmuller空间，包括封闭和开放。曲面的Teichmuller空间是曲面所有可能形状的空间，其中曲面的形状是曲面上的双曲度规，直到等距与恒等同构。因此，曲面上双曲度量的不变量被用于研究Teichmuller空间。π吗?首先考虑双曲平面上的Teichmuller空间，称为全称Teichmuller空间，因为这个空间包含了所有其他的Teichmuller空间。利用双曲平面理想三角剖分的剪切不变量来参数化通用的Teichmuller空间。PI打算继续他的研究一般的Teichmuller空间和相关的Teichmuller空间的这些不变量称为剪切。特别是，PI打算描述有限穿孔封闭表面的Teichmuller空间上的Weil-Petersson度规在表面的理想三角剖分上的剪切量，其中三角剖分既可以是局部有限的，也可以是局部无限的。他还打算在皇后学院的一些本科生和硕士生的帮助下，在计算机上实现这些公式。将剪切不变量应用于Teichmuller空间的另一个方向是找到Takhtajan-Teo Teichmuller空间的剪切参数化，并找到该空间中Weil-Petersson度规的公式。这个方向可能适用于莎朗-芒福德？计算机视觉中的二维形状分析方法。封闭曲面的拟福氏空间也支持Weil-Petersson度规，该度规是通过取极限拟圆的Hausdorff维数与Sullivan-Paterson测度乘积的二阶偏导数来定义的。PI打算研究无穷小的Sullivan-Paterson测度及其交点数，以获得类似于Fuchsian （Teichmuller）空间的情形的准Fuchsian空间上Weil-Petersson测度的另一种表达式。黎曼曲面是二维物体，局部看起来像平面的开放子集，并且具有保持角度的过渡映射。每一个黎曼曲面在它的共形度量类中都支持一个唯一的双曲度量。PI研究曲面上的双曲度规的变化，这个曲面被认为是一个称为Teichmuller空间的度量空间。Teichmuller空间在复杂分析、低维拓扑、动力学、微分几何和物理学中都很有意义。该项目的一个方面与Sharon-Mumford的方法给出的计算机视觉以及Nag-Sullivan和Takhtajan-Teo的方法中的数学物理有关。该项目正在构建通用Teichmuller空间的研究工具，它具有应用于上述领域的潜力。该项目的另一部分涉及皇后学院的本科生和硕士生。参与该项目的学生将接触到积极的研究议程，从而为科学和工程领域的人力资源开发做出贡献。

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.