REU Site: Combinatorics, Probability, and Algebraic Coding Theory

该项目是东田纳西州立大学（ETSU）和波多黎各大学庞塞分校（upp）的合作项目。PI和副PI每三年带领10名本科生进行一系列开放式数学研究项目。每年分配12个项目，其中至少一半的调查结果是经过评审的研究出版物。参与者经过全国范围的筛选，每年前往联合数学会议展示他们的研究成果。为学生选择的项目是当代的；困难的，容易处理的；更广泛的数学界感兴趣的；通常每个新的结果都会引发更多的问题。学生被要求展示他们的作品，并努力发表。因此，他们对研究的介绍与他们在其余职业生涯中所遇到的相似。学生的研究领域由NSF支持的主要研究人员积极调查。从人口统计学上看，该校的学生与ETSU和upp等地方学校的学生相似，反映了我国数学人才的多样性：入选学生中至少有50%是女性；至少40%来自代表性不足的群体；至少40%的人是第一代大学生；至少50%的学生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”（NSF 13-542）。学生研究编码理论、组合学或图论中的极值、代数或概率问题。具体来说，问题要么是代数编码理论，要么是极值/概率离散数学。例如，这两个小组的研究小组研究代数图结构或排列和单词模式的不同方面。PI建议学生使用(a)离散组合概率的深度方法，以及(b)经典组合学-与经典分析（不等式，渐近分析等）相结合。度量集中的概念是其中几个活动的智力焦点。另一方面，由联合项目负责人指导的学生则需要使用来自前沿研究的线性代码。他们将研究来自里德-所罗门码、厄米码和其他代数函数域的码，而其他人则有望为特定应用找到明确的码结构。学生可以从图中确定好的长代码，设计好的编码和解码算法，或者使用格罗布纳基、有限域和有限域上的多项式来找到极值组合对象的明确结构。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 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.