Exploring Dynamical Systems: Exposing Students to Contemporary Mathematics

探索动力系统：让学生接触当代数学

• 批准号: 0205779
• 负责人: Robert Devaney
• 金额: $ 30.43万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-05-15 至 2007-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0205779&HistoricalAwards=false
• 关键词: Exploring; Dynamical; Systems; Exposing; Students

One of the major difficulties facing mathematics today is the fact that most people have no idea about what is happening in contemporary mathematics or why this field is important. Unfortunately, most contemporary research in this field is difficult, if not impossible, to explain to students. The problem is that much of today's mathematics builds upon years or even centuries of prior mathematical research. As a consequence, undergraduate students rarely have the background to experience what is new and exciting in this field.In this project, the PI and his collaborators are attempting to change this culture by creating a series on on-line activities that provide students with the background and the tools necessary to experience and investigate certain contemporary topics in mathematics. The activities are drawn from the areas of mathematics known as dynamical systems theory and fractal geometry. These are two relatively new areas of mathematics that have widespread applicability in all areas of science and engineering. In addition, much of the current research in this field involves a significant experimental component involving computer graphics. Most importantly, much of the PI's current research in these areas involves iteration of simple expressions (such as quadratic functions), and so many of the ideas in this field can be readily explained to students with only a high school background in mathematics. In this project, the PI is developing a series of ten stand-alone modules, each of which is designed to acquaint the user with a particular topic of interest in these fields. Each module includes streaming video introductions, online software (java applets), computer generated animations, and suggested areas for student investigation. The materials being developed can be woven into courses throughout the curriculum, ranging from introductory math courses for liberal arts students and precalculus to calculus and linear algebra courses, to give students an ongoing glimpse of the excitement of modern mathematics. Alternatively, the modules can be concatenated in a string of activities designed to bring the student to the forefront of active research in this area.

当今数学面临的主要困难之一是，大多数人不知道当代数学正在发生什么，也不知道为什么这个领域很重要。不幸的是，该领域的大多数当代研究很难（如果不是不可能的话）向学生解释。问题在于，当今的数学大部分都建立在数年甚至数世纪的先前数学研究的基础上。因此，本科生很少有背景去体验这个领域的新鲜和令人兴奋的事物。在这个项目中，PI 和他的合作者试图通过创建一系列在线活动来改变这种文化，这些活动为学生提供体验和研究某些当代数学主题所需的背景和工具。这些活动来自被称为动力系统理论和分形几何的数学领域。这是两个相对较新的数学领域，在科学和工程的所有领域都有广泛的适用性。此外，当前该领域的许多研究都涉及涉及计算机图形学的重要实验部分。最重要的是，PI 目前在这些领域的研究大多涉及简单表达式（例如二次函数）的迭代，因此该领域的许多想法可以很容易地向只有高中数学背景的学生解释。在这个项目中，PI 正在开发一系列十个独立模块，每个模块都旨在让用户熟悉这些领域中感兴趣的特定主题。每个模块包括流媒体视频介绍、在线软件（java 小程序）、计算机生成的动画以及供学生研究的建议领域。正在开发的材料可以融入整个课程中，从文科学生的数学入门课程和微积分基础课程到微积分和线性代数课程，让学生持续了解现代数学的魅力。或者，这些模块可以串联在一系列活动中，旨在将学生带到该领域积极研究的前沿。