Mathematical Sciences: An Experimental Tool for Topological Surface Dynamics

研究人员开发了一个计算机程序来研究被刺破表面图的拓扑性质。软件工具的一个核心组件是用于Thurston-Nielsen表面同态分类的Bestvina-Handel算法，它不仅产生了分类，而且还产生了同态所属的同位素类的典型代表-“火车轨道图”。这一领域的实验对于定理和猜想的表述是必不可少的，但目前由于对表面同胚的一个例子进行分类和分析所需要的大量劳动而受到阻碍。该软件消除了这个障碍。它允许研究人员快速生成火车轨道地图，以及其悬架的表示和特征（代数和图形）。该软件通过万维网提供。一个是研究动力系统来解释和预测大量的现象，从恒星和行星的运动，到地球大气的循环，到钻机和人类心脏等机械的行为，再到细菌的自我推进。在描述这种具有重复特征的现象时，从某个空间到自身的迭代地图形式的模型往往具有很强的解释和预测能力。也许最简单的一类现实模型是二维曲面的连续映射。即使是这一类的简单成员也会表现出令人困惑的行为复杂性，关于这些系统的许多问题尚未得到回答。穿孔表面的图之所以引起人们的兴趣，是因为它们与周期轨道之间的关系以及结理论中长期存在的问题有关。本计画的目的是开发一个软体工具，用于实验刺穿表面地图的拓扑特性。拓扑性质，即那些在表面或图的扭曲下持续存在的性质，是最基本的。该软件通过万维网提供。

9626884 Menasco The investigators develop a computer program for studying topological properties of maps of punctured surfaces. A central component of the software tool is the Bestvina-Handel algorithm for the Thurston-Nielsen classification of surface homoeomorphisms, which yields not only the classification, but also a canonical representative - a "train-track map" - of the isotopy class to which the homeomorphism belongs. Experimentation in this field, essential to the formulation of theorems and conjectures, is currently hampered by the very large amount of labor entailed in classifying and analyzing even a single example of a surface homeomorphism. The software removes this roadblock. It allows the researcher to rapidly generate train-track maps, and representations and characterizations (both algebraic and graphical) of their suspensions. The software is made available via the World Wide Web. One studies dynamical systems to explain and make predictions about a vast range of phenomena, from the motions of the stars and planets, to the circulation of the Earth's atmosphere, to the behavior of machinery such as drilling rigs and human hearts, to the self-propulsion of bacteria. In describing such phenomena, which are repetitive in character, models in the form of iterated maps from some space to itself frequently have great explanatory and predictive power. Perhaps the simplest class of realistic models is that of continuous maps of two-dimensional surfaces. Even simple members of this class can exhibit a bewildering complexity of behavior, and many questions about these systems have yet to be answered. Maps of punctured surfaces are of interest because of their bearing on the relationships among periodic orbits and on long-standing problems of the theory of knots. The purpose of this project is to develop a software tool for experimentation in the topological properties of maps of punctured surfaces. Topological properties, those that ar e persistent under distortions of the surface or map, are the most fundamental. The software is made available via the World Wide Web.