REU Site: Combinatorics and Coding Theory in the Tropics

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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.