I am an assistant professor of mathematics at Hope College. This fall I'm teaching Calculus 1 and Algebra. In addition to teaching I enjoy working on our departmental newsletter Off on a Tangent. My research interests are in algebraic topology and algebra. The past few summers I mentored students in the Hope College mathematics REU program, and in addition to doing some really interesting math, my students and I had a great time! This past summer my students, Sarah Cobb from Wheaton College in Illinois and Josh Kinder from Hope, and I developed a method for determining all the irreducible representations of wreath products of cyclic groups by using necklaces and their symmetries. Information about my 2009 REU project appears below. Besides mathematics, I enjoy music, hiking, reading and sports, especially baseball.
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2009 REU Project
Students in my research group will investigate questions at the interface of algebra and topology. The exact nature of the project may be more algebraic or more topological depending on interest, but typically the problems we will explore will be algebraic questions whose answers shed light on questions in topology.
Recent REU research projects have focused on representation theory. A representation of a group produces matrices that correspond to the elements of the group and obey the same relations as the group. Representation theory thus allows one to use the tools of linear algebra to study groups. Representation theory has many interesting applications; for instance, representation theory is used to calculate molecular bonding orbitals and allowed vibrational modes of molecules.
Some of these representations are basic in the sense that any representation of the group is comprised of these basic representations, known as irreducible representations. The irreducible representations are thus analogous to prime numbers: just as any number may be decomposed into its unique product of primes, any representation of a group may be decomposed into its irreducible representations. It is well known that the number of irreducible representations of a group equals the number of conjugacy classes in the group, but the exact relationship between the conjugacy classes and irreducible representations is not well understood.
We have developed geometric models for two interesting classes of groups: wreath products of cyclic groups and metacyclic groups. From the model for wreath products, we are able to obtain all the irreducible representations of these groups; from the model for metacyclic groups, we are able to obtain many, but not all, of the irreducible representations of these groups.
There are several unanswered questions in this area that students might explore this summer. First, can the model for metacyclic groups be modified in some way to account for all irreducible representations? Second, can these models be extended to other classes of groups? Third, can the models be used to establish a correspondence between conjugacy classes and irreducible representations? Fourth, how much structure do these geometric models incorporate? For example, how much structure in the geometric model is preserved by Adams operations on the representation ring, for instance?
Background needed: a semester of linear algebra and a semester of abstract algebra; additional coursework in algebra and familiarity with computer algebra systems such as Maple are beneficial, but not necessary.