REU Site: Combinatorics, Probability, and Algebraic Coding Theory

该项目是东田纳西州立大学(ETSU)和庞塞波多黎各大学(UPRP)的合作项目。PI和共同PI在三年中每年带领十名本科生完成一系列开放式数学研究项目。每年有12个项目被分配，调查的结果是至少有一半的案例会发表一篇经过评审的研究论文。参与者是在全国范围内搜索后挑选出来的，并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；困难但容易处理；更广泛的数学界感兴趣；通常每一个新的结果都会带来更多的问题。学生们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接下来的职业生涯中将遇到的情况相似。由国家科学基金会支持的领先研究人员积极调查学生的研究领域。在人口结构上，学生与ETSU和UPRP等地区学校的学生相似，反映了我们国家数学池的多样性：被选中的学生至少50%是女性；至少40%来自代表性较低的群体；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，并被带到“从相对独立的状态到他们的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。具体地说，问题要么在代数编码理论中，要么在极端/概率离散数学中。例如，这两个小组的研究团队致力于代数图结构或排列和单词模式的不同方面的工作。在PI的指导下，学生们使用(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)。集中测量的概念是其中几个活动的智力焦点。另一方面，在联合PI的建议下，学生们预计将使用来自尖端研究的线性代码。他们将使用从里德-所罗门码、厄米特码和来自代数函数域的其他码派生的代码，而其他人将被期望找到用于特定应用的代码的显式构造。学生可以从图中确定好的长码，设计好的编码和解码算法，或者使用有限域上的Grobner基、有限域和多项式来寻找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优势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.