REU Site: Research Challenges of Computational Methods in Discrete Mathematics

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。REU网站“离散数学计算方法的研究挑战”是摩拉维亚大学、雪松Crest学院和库茨敦大学之间的合作项目，每年夏天将有10名学生到摩拉维亚学院学习9周，专注于与离散计算数学相关的研究项目及其在刺激新研究中的作用。该项目的目标是在学生骨干和早期职业研究人员中宣传离散数学计算方法日益增长的重要性，以帮助将其建立为他们研究库中的工具，鼓励参与者继续他们的研究生院教育并追求研究事业，并为培养21世纪的劳动力，特别是女性，少数民族和第一代学生做出贡献。本研究的智力焦点集中在将计算机技术集成到离散计算方法中的重要性日益增加，以及它对其他领域的贡献，例如计算机科学、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院校，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为学生提供很好的服务，包括专业发展研讨会和到当地公司的实地考察，这些活动可以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季桥项目与当地初高中学生互动，并在暑期项目结束时通过地区会议与附近reu的学生交流研究成果。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所无法企及的。在离散数学研究中，计算机模拟在计算机系统的应用中起着特别重要的作用，利用计算机的能力来促进数学发现中新范式的使用。例子包括密码学中的错误检查和连通图上的策略数学游戏等应用，这两者都受益于计算机生成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（以及缩小）猜想，统计显着结果和正式证明之间的差距。研究项目包括离散数学主题，如算术结构及其关键群，宽和拉丁非重复连续分区，图形Nim的计算策略，网络可靠性参数，以及从推广在线整数序列百科全书中现有序列产生的问题。导师有接触、研究计算方法和离散数学开放问题的经验。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.