REU Site: Combinatorics, Probability, and Algebraic Coding Theory

该项目是东田纳西州立大学（ETSU）和庞塞的波多黎各大学（UPRP）之间的合作。PI和合作PI导致十名本科生在每三年的一系列开放式数学研究项目。每年分配12个项目，调查结果至少有一半是经过审查的研究出版物。参与者是在全国范围内搜索后选出的，每年参加联合数学会议以展示他们的研究。为学生选择的项目是当代的;困难但易于处理;更广泛的数学界感兴趣;并且通常会导致更多的问题与每一个新的结果。学生们被要求展示他们的工作，他们奋进发表它。因此，他们对研究的介绍与他们在职业生涯的其余部分将遇到的相似。学生的研究领域由NSF支持的领先研究人员积极调查。从人口统计学上看，学生与ETSU和UPRP等地区学校的学生相似，反映了我国数学人才库的多样性：选定的学生至少50%是女性;至少40%来自代表性不足的群体;至少40%是第一代大学生;至少50%来自本科研究机会有限的学校。学生们得到精心的指导，并采取“从一个相对依赖的地位，作为独立的地位，他们的能力保证”（NSF 13-542）。学生的工作在编码理论，组合数学，或图论的极值，代数，或概率问题。具体来说，问题要么在代数编码理论，或极值/概率离散数学。例如，这两个小组的研究团队致力于代数图构造的各个方面，或排列和单词模式。由PI指导的学生使用（a）离散组合概率的深入方法，以及（B）经典组合数学-与经典分析（不等式，渐近分析等）相结合。措施集中的概念是其中几项活动的智力焦点。另一方面，由co-PI建议的学生预计将使用来自前沿研究的线性代码。他们将与来自Reed-Solomon码，Hermitian码和代数函数域的其他代码的代码一起工作，而其他人将有望找到特定应用的代码的显式构造。学生可以从图形中确定好的长代码并设计好的编码和解码算法，或者使用Grobner基、有限域和有限域上的多项式来找到极值组合对象的显式构造。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而被认为值得支持。

The project is a collaboration between East Tennessee State University (ETSU) and the University of Puerto Rico in Ponce (UPRP). The PI and the co-PI lead ten undergraduates during each of three years in a series of open ended mathematical research projects. Twelve projects are assigned each year, and the investigations result in a refereed research publication in at least half the cases. Participants are selected after a nationwide search, and travel to the Joint Mathematics Meetings each year to present their research. Projects chosen for the students are contemporary; difficult but tractable; of interest to the wider mathematical community; and usually lead to more questions with every new result. Students are required to present their work and they endeavor to publish it. Their introduction to research is thus similar to what they will encounter during the rest of their careers. The students' areas of research are actively investigated by leading researchers supported by NSF. Demographically, students are similar to those at regional schools such as ETSU and UPRP, and reflect the diversity of our nation's mathematical pool: Selected students are least 50% female; at least 40% are from underrepresented groups; at least 40% are first generation college attendees; and at least 50% are 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" (NSF 13-542).Students work on extremal, algebraic, or probabilistic problems in coding theory, combinatorics, or graph theory. Specifically, problems are either in algebraic coding theory, or extremal/probabilistic discrete mathematics. For example, research teams in these two groups work on diverse aspects of algebraic graph constructions, or permutation and word patterns. Students advised by the PI 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 is the intellectual focus of several of these activities. The students advised by the co-PI, on the other hand, are expected to work with linear codes from cutting-edge research. They will work with codes derived from Reed-Solomon codes, Hermitian codes and other codes from algebraic function fields, while others will be expected to find explicit constructions of codes for specific applications. Students might determine good, long codes from graphs and design good encoding and decoding algorithms, or use Grobner bases, finite fields, and polynomials over finite fields in order to find explicit constructions of extremal combinatorial objects.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.