REU Site: Research Challenges of Computational Methods in Discrete Mathematics

该奖项的全部或部分资金根据《2021 年美国救援计划法案》（公法 117-2）提供。 REU 网站“离散数学计算方法的研究挑战”是摩拉维亚大学、锡达克雷斯特学院和库兹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九周，专注于与离散计算数学及其在刺激新研究中的作用相关的研究项目。该计划的目标是向一批学生和早期职业研究人员宣传离散数学中计算方法日益增长的重要性，以帮助将其建立为研究工具库中的工具，鼓励参与者继续进入研究生院学习并从事研究职业，并为二十一世纪的劳动力特别是女性、少数族裔和第一代学生的培训做出贡献。这项研究的智力重点集中在将计算机技术集成到离散计算方法中的日益重要，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机构，重点是招募不同的学生群体。学生研究项目之外计划的活动也将为学生提供良好的服务，包括专业发展研讨会和当地公司实地考察，这些活动可以增加参与者接触职业机会的机会。参与者还将通过 Lehigh Valley Summerbridge 项目与当地初中和高中学生互动，并在夏季项目结束时通过区域会议与附近 REU 的学生互动，分享他们的研究成果。现代计算机技术使得以前几代数学家无法达到的计算速度和精度探索数学概念成为可能。计算机模拟在计算机系统用于离散数学研究的应用中尤其发挥着核心作用，利用计算机的力量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策略的数学游戏等应用，这两者都受益于计算机生成的示例产生的洞察力。该计划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，例如算术结构及其临界群、宽拉丁文非重复连续分区、图形 Nim 的计算策略、网络可靠性参数以及概括在线整数序列百科全书中现有序列所产生的问题。导师们在计算方法和离散数学中的开放问题的获取、研究和贡献方面拥有丰富的经验。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力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.