General Research Interests
I am currently working on classifying the subloops of Zorn Vector Matrix algebras over commutative rings. It has been known for some time that these loops are Moufang, and there has been a lot of work done with the classic construction over finite fields. I want to see if different loops can be obtained over commutative rings, or if there is a natural way to classify these loops in terms of those over fields. Such a classification may help find new non-associative Moufang loops or provide a different way of organizing the known ones.
I also am very interested in the question of which Moufang Loops can be expressed as Zorn Vector Matrices over a Ring. I'd like to develop a kind of representation theory for loops in this way. This may also raise a new insightful way to organize the known Moufang Loops, ease some Moufang Loop computations, and could potentially be a valuable field in itself.
Talks Given
"Moufang Loops Arising from Zorn Vector Matrix Algebras" October 18 2008, AMS Fall Central Section Meeting, Western Michigan University, Kalamazoo, MI. Meeting #1043, Notices Program Issue: October 2008, Abstract Issue: 29/4. Special Session on Quasigroups and Loops.
Past Projects
In the Summer of 2003, I participated in an REU program at Hope College. I worked under Dr. Darren Stephenson on a project involving finding certain quadratic forms which had applications to Clifford Algebras.
In the summer of 2004, I participated in an REU program at SUNY Potsdam. I worked under Dr. Harold Ellingsen on a project involving classifying the groups of units of quaternion algebras over integer rings.