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位于庞塞市的波多黎各大学、威廉姆斯学院和东田纳西州立大学的联合项目。教师导师将领导一系列开放式数学研究项目，十名本科生参与者和两名经验丰富的同行导师每年夏天都会参与，为期三年，为期九周。为学生选择的研究项目是当代的；困难但容易处理的；更广泛的数学界感兴趣的；由NSF支持的领先研究人员积极调查的研究领域；通常每个新结果都会引发更多问题。虽然这些调查通常会在同行评议的期刊上发表文章，但该网站提供的体验类似于学生在研究生涯中遇到的体验。从人口结构上看，参与者反映了美国劳动力市场的多样性：入选的学生群体至少有50%是女性；至少40%来自代表性不足的群体；至少40%是第一代大学学生；至少50%来自本科研究机会有限的学校。学生被仔细地指导和引导，“从一个相对独立的地位转变为他们的能力所保证的独立地位”(Nsf19852)。该项目在波多黎各的庞塞举行；提供双语研究环境；并从美国的拉美裔和波多黎各的拉美裔建立了丰富的拉美裔研究社区，为拉美裔学生创造了熟悉的环境，并为波多黎各学生在美国大陆的研究生学习生活做好了准备。REU拥有经验丰富的同行导师，他们已经成功地处理了REU以前的研究项目，并成功地解决了本科学习中的困难。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--结合经典分析(不等式、渐近分析等)。集中测量的概念仍然是学生几项研究活动的智力焦点。还提供了停车功能组合方面的项目。这些课程从经典的计数技术开始，根据学生的背景和兴趣，进一步探索与超平面排列、偏序集、Young图面和离散几何有关的问题。学生将使用编程来发现模式，并制定猜想，从而证明该领域的结果。在与编码理论相关的专题中，学生可能会改进二进制Goppa码不同类别参数的界限，可能会改进现有的代数几何代码的编码和解码算法，找到Polar Grassmann码参数的界限，或者将线性码应用于DNA条形码。该奖项反映了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。