Mathematical Sciences: An Experimental Tool for Topological Surface Dynamics

数学科学：拓扑表面动力学的实验工具

• 批准号: 9626884
• 负责人: William Menasco
• 金额: $ 9.2万
• 依托单位: SUNY at Buffalo
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626884&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Experimental; Tool; Topological

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.

小行星9626884 研究人员开发了一个计算机程序，用于研究穿孔表面映射的拓扑性质。 该软件工具的一个核心组成部分是用于表面同胚的Thurston-Nielsen分类的Bestvina-Handel算法，该算法不仅产生分类，而且产生同胚所属的合痕类的典型代表-“火车轨道图”。 在这一领域的实验，必不可少的公式化的定理和图解，目前阻碍了非常大量的劳动所需的分类和分析，即使是一个单一的例子，表面同胚。 软件消除了这个障碍。 它允许研究人员快速生成列车轨道地图，以及其悬挂的表示和表征（代数和图形）。 该软件通过万维网提供。 一个研究动力系统来解释和预测各种现象，从恒星和行星的运动，到地球大气的循环，到机械的行为，如钻机和人类心脏，到细菌的自我推进。 在描述这些具有重复性的现象时，从某个空间到自身的迭代映射形式的模型往往具有很强的解释力和预测力。 也许最简单的一类真实模型是二维表面的连续映射。 即使是这个类的简单成员也可以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解答。 穿孔曲面的映射之所以引起人们的兴趣，是因为它们关系到周期轨道之间的关系和纽结理论中长期存在的问题。 这个项目的目的是开发一个软件工具，用于实验穿孔表面的地图的拓扑特性。 拓扑性质是最基本的，这些性质在表面或地图的扭曲下是持久的。 该软件通过万维网提供。