Geometry and applications of deformations of Riemann surfaces

调查员将继续他的研究测地线（最短路径），凸性和曲率（基本描述几何）的威尔-彼得森几何Teichmueller空间。研究者已经证明了Teichmueller空间是一个无限多面体，有无限多个顶点和测地线面以直角相交。 描述给出了一个简短的证明的Masur-Wolf结果确定充分对称群的多面体。调查人员提供了关于面部附近几何形状的详细信息。双曲曲面上的闭测地线（最短闭路）的长度是用于在Teichmueller空间中定位点的参数。研究人员将继续他的计划，使用分析的变化的长度来描述合成和微分几何的Teichmueller空间。 这种描述比研究者先前的方法更简单，目前许多研究人员正在使用这种方法。研究者正在根据长度的变化对曲率进行描述。 研究人员还将研究谐波（最小总平方能量）映射到无限多面体和从无限多面体。他还将继续研究Weil-Petersson测地动力系统，以及黎曼曲面的模空间的相交理论。黎曼曲面是一个二维曲面，每个点都有角度测度的概念。一个黎曼曲面至少有两个手柄被赋予非欧（双曲）几何确定距离，最短路径和振动膜运营商。黎曼曲面有各种形状，由厚和薄子区域的相对位置描述。Teichmueller空间是具有给定数量的把手的黎曼曲面的所有可能形状的空间。单个黎曼曲面的双曲几何导致Teichmueller空间的Weil-Petersson几何。黎曼曲面为振动膜、传播粒子和分形提供了模型。

The investigator will continue his study of geodesics (shortest paths), convexity and curvature (the fundamental descriptor for a geometry) for the Weil-Petersson geometry for Teichmueller space. The investigator has already shown that Teichmueller space is an infinite polyhedron with an infinite number of vertices and geodesic faces meeting at right angles. The description gives a short proof of the Masur-Wolf result determining the full symmetry group of the polyhedron. The investigator has provided detailed information on the geometry near the faces. Lengths of closed geodesics (shortest closed paths) on a hyperbolic surface are parameters for locating a point in Teichmueller space. The investigator will continue his program of using analysis of variations of the lengths to describe the synthetic and differential geometry of Teichmueller space. The description is simpler than the investigator's prior approach, which is presently used by a number of researchers. The investigator is developing a description of curvature in terms of variations of lengths. The investigator will also study harmonic (least total square energy) mappings to and from the infinite polyhedron. He will also continue to study the Weil-Petersson geodesic dynamical system, as well as intersection theory for the moduli space of Riemann surfaces.A Riemann surface is a two-dimensional surface with a notion of angle measure at each point. A Riemann surface with at least two handles is endowed with non Euclidean (hyperbolic) geometry determining distance, shortest paths and a vibrating membrane operator. Riemann surfaces come in various shapes, described by the relative locations of thick and thin subregions. Teichmueller space is the space of all possible shapes for a Riemann surface with a given number of handles. The hyperbolic geometry of individual Riemann surfaces leads to the Weil-Petersson geometry for Teichmueller space. Riemann surfaces provide models for vibrating membranes, propagating particles and fractals.