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的联合项目。教师导师将带领一系列开放式数学研究项目10名本科生参与者和两名经验丰富的同行导师，为期三年，每年夏天为期九周。为学生选择的研究项目是当代的；困难的，容易处理的；更广泛的数学界感兴趣的；在NSF支持的主要研究人员积极研究的领域；通常每个新的结果都会引发更多的问题。虽然这些调查通常会在同行评议的期刊上发表，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。从人口统计学上看，参与者反映了美国劳动力的多样性：入选的学生群体中至少有50%是女性；至少40%来自代表性不足的群体；至少40%的第一代大学生；至少50%来自本科生研究机会有限的学校。学生们受到精心的指导，并被“从相对依赖的状态转变为他们能力所保证的独立状态”（nsf19852）。该项目在波多黎各的庞塞市进行；提供双语研究环境；并建立了一个丰富的西班牙裔研究社区，包括美国的拉丁裔和波多黎各的拉丁裔，为西班牙裔学生创造了一个熟悉的环境，为波多黎各的学生在美国大陆的研究生学习生活做好准备。REU有很强的同伴指导方面，由经验丰富的同伴导师提供，他们已经成功地处理了以前REU的研究项目，并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、离散几何和经典组合。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-与经典分析（不等式，渐近分析等）相结合。测度集中的概念仍然是几个学生研究活动的智力焦点。还提供了停车功能组合的项目。这些课程从经典的枚举技术开始，并基于学生的背景和兴趣，进一步探讨与超平面排列、部分有序集、杨氏表和离散几何相关的问题。学生将使用编程来揭示模式并形成导致该领域结果证明的猜想。在编码理论相关的项目中，学生可以改进不同类型的二进制Goppa码的参数界，可以改进现有的代数几何码的编码和解码算法，可以找到Polar Grassmann码的参数界，或者将线性码应用于DNA条形码。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。

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.