Zunfeng "Jeff" Chen '08 (Brooklyn, N.Y.) and Jonathan Stults '07 (Woodstock, N.Y.) are at Hamilton for summer research into mathematics. They have abandoned the topic they originally chose ("difference equations, differential equations, and Simpson's paradox") and moved to the study of n by n (square) matrices. "We're counting all n by n matrices in Z mod p with all or no eigenvalues in Z mod p."
A matrix is a representation of a transformation; it also allows you to transform vectors into other vectors. Many computerized systems run off matrices; Google uses them to sort data. The set is interesting in mathematical terms because of unique properties which make it useful in internet security.
Chen and Stults are working with a specific variety of matrices called modular matrices. For these, the entries are in the set Z mod p where p is any prime number (this is a set which contains the numbers 0 through p-1). They are looking at a specific set of matrices: those whose eigenvalues are also in Z mod p, or those matrices with no eigenvalues.
An eigenvector is a vector which does not rotate when transformed, but is simply elongated or truncated by a scalar: its associated eigenvalue. Matrices are usually associated with a characteristic polynomial and the eigenvalues are the roots of these polynomials. Chen and Stults are using the characteristic polynomials of their matrices, and how these polynomials change when the matrix changes, to study eigenvalues. They are also trying to extend their 2x2 case to a more general nxn case.
Stults uses Group Theory, a way of "breaking down a [finite] group or set," to count his matrices. Chen is addressing the same issue, but in a different way; he has developed a different way of counting which bypasses Group Theory. He explains that his approach to counting matrices is simpler, but the actual way of counting is harder. Nearing the end of their summer, Chen and Stults are examining their matrices in terms of polynomials, not Group Theory. They have found many properties which seem unrelated, but recently these properties appear to be linked after all. Their "seemingly random properties" are dove-tailing. They have also proved something which is universally true for matrices, and extended their 2x2 case into a 3x3 case.
This is the first summer of research for both students. "I'm having a good time," says Stults. "I get paid to think about math all day." Although at times things are a bit tricky, they are both enjoying themselves. "You have to break it down into nice pieces," Chen comments on the hard parts of the project, "you have to partition them."
Stults is a TA for Linear Algebra, a grader, a peer tutor, and a tutor in the Quantitative Literacy Center. Chen is a tutor for HEOP and math, and plays table tennis. They are advised in this project by Professor of Mathematics Larry Knop.
- Lisbeth Redfield
