REU Site: Research Challenges of Computational Methods in Discrete Mathematics

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。REU网站“离散数学中计算方法的研究挑战”是由摩拉维亚大学、Cedar Crest学院和库茨敦大学合作建立的，每年夏天将有10名学生到摩拉维亚学院学习9周，集中研究与离散计算数学相关的研究项目及其在激发新研究方面的作用。该计划的目标是在一批学生和职业生涯早期的研究人员中宣传离散数学计算方法日益增长的重要性，帮助将其确立为他们研究武器库中的工具，鼓励参与者继续他们的研究生教育并从事研究工作，并为培训21世纪的劳动力，特别是在女性、少数族裔和第一代学生中做出贡献。这项研究的智力焦点集中在将计算机技术整合到离散计算方法中的日益重要的意义，以及它对其他领域的贡献，如计算机科学、生物学、物理学、数据安全和医学中的算法应用。导师来自宾夕法尼亚州利哈伊谷地区的机构，重点将放在招收不同群体的学生上。学生研究项目之外的计划活动也将很好地服务于学生，包括专业发展研讨会和到当地公司的实地考察，这可能会增加参与者对职业可能性的接触。参与者还将通过利哈伊谷夏桥项目与当地初中生和高中生互动，以及通过暑期项目结束时的地区会议与附近雷乌斯的学生互动，分享他们的研究成果。现代计算机技术使探索数学概念成为可能，其计算速度和精度是前几代数学家无法企及的。特别是计算机模拟在应用计算机系统进行离散数学研究方面发挥着核心作用，利用计算机的能力来促进在数学发现中使用新的范例。例如密码学中的错误检查和连通图上的策略数学游戏等应用，这两个应用都受益于计算机生成的示例所产生的洞察力。本课程将为学生提供必要的工具来理解(以及弥合)猜想、统计学上有意义的结果和正式证明之间的差距。研究项目包括离散数学主题，如算术结构及其临界组，宽的和拉丁的非重复连续划分，图形NIM的计算策略，网络可靠性参数，以及在在线整数序列百科全书中推广现有序列所产生的问题。导师们有接触、研究计算方法和离散数学公开问题的经验。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). The REU site “Research Challenges of Computational Methods in Discrete Mathematics” is a collaboration between Moravian University, Cedar Crest College, and Kutztown University that will bring ten students to Moravian College for nine weeks each summer to concentrate on research projects associated with discrete computational mathematics and its role in stimulating new research. The program goals are to publicize the growing importance of computational methods in discrete mathematics among a cadre of students and early-career researchers to help establish it as a tool in their research arsenal, encourage participants to continue their education into graduate school and pursue careers in research, and contribute toward the training of the twenty-first century workforce particularly among women, minorities, and first-generation students. The intellectual focus of this research concentrates on the increasing importance of integrating computer technology into discrete computational methodology, as well as its contribution to other fields, such as algorithmic applications in computer science, biology, physics, data security, and medicine. Mentors are drawn from institutions within the Lehigh Valley region in Pennsylvania, and an emphasis will be placed on recruiting diverse groups of students. The planned activities outside of the student research projects will also serve the students well, including professional development workshops and field trips to local companies that could increase participants’ exposure to career possibilities. Participants will also interact with local middle and high school students through the Lehigh Valley Summerbridge program, as well as with students in nearby REUs through a regional conference at the end of the summer program to share their research. Modern computer technology makes it possible to explore mathematical concepts with computational speed and precision not available to previous generations of mathematicians. Computer simulation in particular plays a central role in the application of computer systems for research in discrete mathematics, harnessing computer power to promote the use of new paradigms in mathematical discovery. Examples include applications such as error checking in cryptography and mathematical games of strategy on connected graphs, both of which benefit from the insight produced by computer-generated examples. This program will equip students with the necessary tools to understand (as well as close) the gaps between conjecture, a statistically significant result, and a formal proof. Research projects include discrete mathematics topics such as arithmetic structures and their critical groups, wide and Latin non-repeating consecutive partitions, computing strategies for graphical Nim, network reliability parameters, and problems emanating from generalizing existing sequences in the Online Encyclopedia of Integer Sequences. The mentors have experience accessing, researching, and contributing to open problems in computational methods and discrete mathematics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.