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

主要研究员研究Teichmuller空间的表面，包括封闭和开放。曲面的Teichmuller空间是曲面的所有可能形状的空间，其中曲面的形状是曲面上的双曲度量，直到等距同伦于单位元。因此曲面上双曲度量的不变量被用于Teichmuller空间的研究。私家侦探？的方法是首先考虑双曲平面的Teichmuller空间，称为泛Teichmuller空间，因为这个空间包含所有其他的Teichmuller空间。一个不变量称为剪切相关的一个理想的三角形的双曲平面是用来参数化的通用Teichmuller空间。PI打算继续他的研究普遍的Teichmuller空间和相关的Teichmuller空间在这些不变量称为剪切。特别地，PI打算用曲面的理想三角剖分上的剪切来描述一个带穿孔闭曲面的Teichmuller空间上的Weil-Petersson度量，其中三角剖分可以是局部有限的，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空间在剪切方面的参数化，并找到该空间中Weil-Petersson度量的公式。这个方向有可能适用于莎伦-芒福德？的方法，在计算机视觉中的二维形状分析。闭曲面的拟Fuchsian空间也支持Weil-Petersson度量，该度量通过取极限拟圆的Hausdorff维数与Sullivan-Paterson测度的乘积的二阶偏导数来定义。PI打算研究无穷小Sullivan-Paterson测度及其相交数，以获得Quasifuchsian空间上Weil-Petersson度量的另一种表达式，类似于Fuchsian（Teichmuller）空间的情况。Riemann曲面是二维对象，局部看起来像平面的开子集，并且具有保持角度的过渡映射。每一个黎曼曲面在其共形度量类中支持一个唯一的双曲度量。PI研究双曲度量在一个被认为是度量的单一空间的表面上的变化，称为Teichmuller空间。Teichmuller空间在复分析、低维拓扑、动力学、微分几何和物理学中很有意义。该项目的一个方面是与计算机视觉的方法给出的沙龙芒福德以及数学物理的方法Nag-Sullivan和Takhtajan-Teo。该项目正在构建用于研究普适Teichmuller空间的工具，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。参加该项目的学生将接触到一个积极的研究议程，从而有助于科学和工程的人力资源开发。

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.