Mathematical Sciences: REU: Experimental Mathematics

在这个面向本科生的研究体验项目中，六名学生将与教师顾问合作，将计算技术的进步与数学研究联系起来。这项工作的重点将放在两个领域的计算方面：群论和动力学。在最初的指导之后，学生们将分成小组，专注于两个主题中的一个。实验和理论之间的相互作用将使学生能够在早期阶段处理手工无法完成的复杂例子。这些将导致对群、动力系统及其底层结构的更好理解。专注于群论的学生将通过置换群和接触Cayley和其他能够进行符号操作的编程语言来介绍计算群论的基本概念。动力系统将建立在学生微积分和线性代数基础课程的基础上，研究轨道的性质，如倍周期和鞍点分叉、Mandelbrot集以及使用迭代函数系统生成分形图像。随着学生越来越熟悉计算机实验，他们将接触到问题区域和未解决的问题，在接下来的练习中，他们将把注意力集中在这些问题上。在研究计划的同时，还将举办一系列晚间科学、历史和文学讲座。

In this Research Experiences for Undergraduates project, six students will work with faculty advisors relating advances in computational technology to mathematical research. The emphasis of the work will be on computational aspects of two areas: group theory and dynamics. Following initial orientation, students will break up into teams, concentrating on one of the two themes. The interplay between experimentation and theory will enable the students to deal, at an early stage, with complicated examples which cannot be done by hand. These will lead to a better understanding of groups, dynamical systems and their underlying structure. Students concentrating on group theory will be introduced to fundamental concepts of computational group theory through permutation groups and exposure to CAYLEY and other programming languages capable of symbolic manipulation. Dynamical systems will build on students' basic courses in calculus and linear algebra to investigate properties of orbits such as period doubling and saddle node bifurcations, Mandelbrot sets and the generation of fractal images using iterated functions systems. As the students become more familiar with computer experimentation, they will be exposed to problem areas and unsolved problems on which they will focus their attention for the remainder of the activity. Interspersed with the research program will be an evening lecture series in the sciences, history and literature.