REU Site: Mathematics Research Experience for Undergraduates

该项目是纽约州立大学波茨坦分校（SUNY Potsdam）与克拉克森大学（Clarkson University）合作，成功开展的为期八周的数学本科生暑期研究经验（REU）的延续。学生每天将以3或4人为一组，与一名顾问一起研究数学中的原创性问题，特别是在代数图论，拓扑图论，分析和代数领域。更具体地说，在该计划的第一年，我们建议一组将研究加性和乘性噪声对微分方程系统稳定性的影响，重点是生物应用系统。另一组将研究具有平面空间嵌入的图形（这种嵌入的每个循环都与一个只沿其边界与图形相交的圆盘相结合）。另一组将研究时间尺度上的动态方程（时间尺度是实数的非空、封闭子集）。特别是，当时间尺度是康托集时，学生将计算泰勒单项式。最后，另一个小组将研究图上的量子行走，这对量子计算和算法有影响。我们将在全国范围内招收完成2年或3年本科课程的数学专业学生。特别鼓励妇女、少数民族、残疾学生和来自研究机会有限的学校的学生申请。我们对学生有远大的目标。我们希望帮助他们建立对自己独立研究能力的信心，并帮助他们建立对广阔数学领域的欣赏和理解。我们也想帮助他们提高口头和书面沟通能力。每个小组将向其他学生展示他们的结果，在八周内至少两次。学生们将被鼓励在全国会议上展示他们的成果。每个小组还将使用LaTex来准备一篇关于他们工作的论文。我们将通过每周一次的客座演讲，以及研究生院的小组讨论，进一步向学生介绍数学专业。大多数数学专业的学生并没有像本科生那样接触到数学研究的过程。我们相信我们的课程为任何严肃的数学专业的教育提供了一个有价值的附加维度。我们相信，他们将带着对数学发现和传播过程更广泛的理解回到自己的母校。我们也会指导我们的学生修改他们的作业，并在会议上发言。我们相信，我们的许多学生将希望继续在研究生院进一步研究数学作为参加我们的计划的结果。少数人会选择不这样做，但他们将带着对数学更深刻的理解和欣赏进入社会。

This project is the continuation of a successful eight-week Summer Research Experience for Undergraduates (REU) in mathematics at SUNY Potsdam, with the cooperation of Clarkson University, in Potsdam, NY. Students will work daily in groups of 3 or 4 with an advisor on an original research problem in mathematics, specifically in the areas of algebraic graph theory, topological graph theory, analysis and algebra. More specifically, in the first year of the program, we propose that one group will study the impact of additive and multiplicative noise on the stability of systems of differential equations with an emphasis on systems that arise from biological applications. Another group will examine graphs that have a flat spatial embedding (an embedding for which every cycle bounds a disk that intersects the graph only along its boundary). Another group will study dynamic equations on time scales (a time scale is a nonempty, closed subset of the real numbers). In particular, students will calculate Taylor monomials when the time scale is the Cantor set. Finally, another group will examine quantum walks on graphs, which has implications for quantum computations and algorithms.We will nationally recruit math majors who have completed 2 or preferably 3 years of undergraduate course-work. Women, minorities, students with disabilities, and students from schools with limited research opportunities will be especially encouraged to apply. We have ambitious goals for our students. We would like to help them build their confidence in their ability to do research independently, and to help them build their appreciation and understanding of the vast field of mathematics. We also want to help them to improve their oral and written communication skills. Each group will present their results to the other students, at least twice during the eight weeks. Students will be encouraged to present their results at a national meeting. Each group will also use LaTex to prepare a paper on their work. We will further introduce our students to the mathematics profession through hosting a weekly guest speaker, and by presenting a panel on graduate school. Most math majors are not exposed to the process of mathematical research as undergraduates. We believe that our program provides a valuable added dimension to any serious math major's education. We believe they will go back to their home institutions with a much broader understanding of the process of mathematical discovery and dissemination. We will also mentor our students as they go through revising their work, and presenting at conferences. We believe that many of our students will want to go on to further study of mathematics in graduate school as a result of participating in our program. A few will choose not to, but they will enter society with a greater understanding and appreciation of mathematics.