REU Site: Combinatorics, Probability, and Algebraic Coding Theory

East Tennessee State University, Johnson City TN

Investigators

Abstract

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.