Exploring Dynamical Systems: Exposing Students to Contemporary Mathematics

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中发生了什么，也不知道为什么这个领域很重要。不幸的是，这个领域的大多数当代研究即使不是不可能，也很难向学生解释清楚。问题是，今天的许多数学是建立在几年甚至几个世纪以前的数学研究之上的。因此，本科生很少有经验的背景，什么是新的和令人兴奋的在这个领域。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。这些活动是从被称为动力系统理论和分形几何的数学领域中提取的。这是数学的两个相对较新的领域，在科学和工程的所有领域都有广泛的适用性。此外，目前该领域的许多研究都涉及计算机图形学的重要实验组成部分。最重要的是，PI目前在这些领域的许多研究都涉及简单表达式的迭代（如二次函数），并且该领域的许多思想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10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 applet），计算机生成的动画，以及建议学生调查的领域。正在开发的材料可以编织到整个课程中，从文科学生的数学入门课程和微积分预备课程到微积分和线性代数课程，让学生不断了解现代数学的兴奋。或者，这些模块可以连接在一系列活动中，旨在将学生带到该领域积极研究的最前沿。

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.