2013 Project 1: Group Testing
Group Testing is a topic that has applications to high-throughput screening and multiplexed probing of DNA clones. Various models are under-studied especially in good designing matrices for small parameters.
2013 Project 2: Tiling of Hadamard Difference Sets
Certain combinatorial configutations (especially, Hadamard matrices and weighing matrices) may be constructed by piecing together certain small structures called ``perfect ternary arrays" (PTAs). In this way, thePTAs create a ``tiling" of the configuration, just as pieces of a jigsaw puzzle create a larger picture. We will examine ``tilings" as special elements of a group ring and attempt to create a general theory of tilings, beginning with primitive (or ``prime") elements and constructing larger tiles from these. Along the way we will be guided by related questions in algebraic number theory and the representation theory of finite groups.
The project has applications to combinatorial designs and digital signal processing and will require a first course in higher algebra.
|Program Director: Vadim Ponomarenko|
|Dr. Ponomarenko has been directing undergraduate and REU research for over a decade. To date, almost every REU participant he has worked with has been coauthor on at least one paper as a result of this collaboration (19 so far, over 7 projects, with 4 more pending).|
|Technical Consultant: Ken W. Smith||Technical Consultant: Robert Ellis|
|In twenty years of involvement with undergraduate students, Dr. Ken Smith has directed a variety of research projects in matrix analysis, cryptography, graph theory and algebraic combinatorics. He has directed undergraduate research as part of an REU program at Central Michigan University (2001-2007), as a technical director for an NSA Summer Program in 1993 (working with approximately 15 undergraduate students on classified research problems in algebra and combinatorics) and as a coordinator for the LURE NSF program (2007-2011.) Whether working as a member of the faculty or as an administrator, professor Smith has promoted research experience for undergraduates. As Director of the Office of Institutional Research at Central Michigan University, he set up a student internship program that trained students in statistical research. Working with math educators at Central Michigan, Smith was co-PI on an NSF grant on "Problem-based, technology-enhanced" (PBTE) preparation of secondary teachers, emphasizing problem-based learning strategies.||Dr. Ellis is an Associate Professor of Applied Mathematics at the Illinois Institute of Technology, and has advised 7 undergraduate research projects since 2003. He has also advised team capstone projects leading to ResearchWeb, a researcher/project matching site, and URJiit, the Undergraduate Research Journal at IIT.|