REU Site: Combinatorics and Coding Theory in the Tropics

REU 网站：热带地区的组合学和编码理论

• 批准号: 2150434
• 负责人: Fernando Pinero
• 金额: $ 43.52万
• 依托单位: University of Puerto Rico in Ponce
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-02-15 至 2026-01-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2150434&HistoricalAwards=false
• 关键词: REU; Site; Combinatorics; Coding; Theory

The REU Site: Combinatorics and Coding Theory in the Tropics is a joint project between the University of Puerto Rico in Ponce, Williams College, and East Tennessee State University. Faculty mentors will lead in a series of open ended mathematical research projects ten undergraduate participants and two experienced peer mentors each summer of three years for nine weeks. The research projects chosen for the students are contemporary; difficult but tractable; of interest to the wider mathematical community; in areas of research actively investigated by leading researchers supported by NSF; and usually lead to more questions with every new result. Whereas the investigations often result in publications in peer reviewed journals, the site provides an experience akin to what students would encounter in a research career. Demographically, participants reflect the diversity of the nation’s workforce pool: The selected student groups are at least 50% female; at least 40% from underrepresented groups; at least 40% first generation college attendees; and at least 50% from schools with limited undergraduate research opportunities. Students are carefully mentored and taken “from a relatively dependent status to as independent a status as their competence warrants” (nsf19852). The program takes place in Ponce, Puerto Rico; offers a bilingual research environment; and builds a rich Hispanic research community from both Latinos in the U.S. and Latinos in Puerto Rico, creating a familiar environment for Hispanic students and preparing students from Puerto Rico for graduate studies life in mainland U.S. The REU has a strong peer mentorship aspect provided by experienced peer mentors who have already been successful in dealing with previous REU’s research projects and in navigating the difficulties in undergraduate studies.Research areas include algebraic coding theory, probabilistic combinatorics, discrete geometry, and classical combinatorics. Students may use (a) deep methods in discrete combinatorial probability, and (b) classical combinatorics -- in tandem with classical analysis (inequalities, asymptotic analysis etc.). The concept of concentration of measure continues to be at the intellectual focus of several of the students' research activities. Projects in the combinatorics of parking functions are also offered. These begin with classical enumerative techniques, and based on student background and interest, further explore problems related to hyperplanes arrangements, partially ordered sets, Young tableaux, and discrete geometry. Students will use programming to uncover patterns and formulate conjectures leading to proofs of results in the area. In projects related to Coding Theory students might improve bounds on parameters of difference classes of binary Goppa codes, might improve on existing encoding and decoding algorithms of algebraic geometry codes, find bounds on the parameters of Polar Grassmann codes, or use linear codes in applications to DNA barcoding.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.

REU 网站：热带地区的组合学和编码理论是波多黎各大学庞塞分校、威廉姆斯学院和东田纳西州立大学教师导师的联合项目，将领导一系列开放式数学研究项目，十名本科生参与者和每年夏天，为期九周的两名经验丰富的同行导师为学生选择的研究项目是当代的，但容易引起更广泛的数学界的兴趣；引发更多问题除了调查结果通常会在同行评审的期刊上发表之外，该网站还提供了类似于学生在研究生涯中遇到的体验。至少 50% 是女性；至少 40% 来自代表性不足的群体；至少 50% 来自本科研究机会有限的学校。其能力所保证的独立地位”（nsf19852），该项目在波多黎各庞塞进行，提供了一个双语研究环境；并建立了一个来自美国拉美裔和波多黎各拉美裔的丰富的研究社区，创造了一个熟悉的社区。 REU 拥有强大的同伴指导能力，由经验丰富的同伴导师提供，他们已经成功处理过 REU 之前的研究项目和研究领域包括代数编码理论、概率组合学、离散几何和经典组合学。学生可以使用（a）离散组合概率的深层方法，以及（b）经典组合学——与经典组合学相结合。分析（不等式、渐近分析等）。测量集中的概念仍然是停车组合学中一些学生研究活动的智力焦点。这些从经典的枚举技术开始，根据学生的背景和兴趣，进一步探索与超平面排列、部分有序集、Young 画面和离散几何相关的问题。学生将使用编程来发现模式并提出猜想。在与编码理论相关的项目中，学生可能会改进二进制 Goppa 代码的不同类参数的界限，可能会改进代数几何代码的现有编码和解码算法，找到以下参数的界限。 Polar Grassmann 代码，或在 DNA 条形码应用中使用线性代码。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。