Interests and Activities

• My research interests. My research interests focus on the interaction of two branches of ring theory.
• Cohn, Schofield, and Bergman explored the techniques of universally inverting maps between projective modules as a way of localizing rings. The nature of the technique does not lead to easy examination of the resulting ring. The resulting rings however are often interesting objects in their own right. My research looks at ways of explicitly producing these rings. (Explicit here means that the elements have normal forms that can be used for computations.) In particular I look at ways of reducing the instruction "invert all full maps" to "invert a well selected finite set of maps".
• One place that this technique shows promise is with finite dimensional algebras. The techniques of universal localization give promise of explicitly producing various large modules, like the generic modules. Most of the work in this field has focused on modules that are finite dimensional over a field that is central to the original ring. The focus seems to be driven by the fact that the homological methods use double duals. The universal localization techniques do not seem to require this finite dimensional restriction.

  Academic interests - My academic interests include:

• Calculus reform - The department is deep into the process of implementing calculus reform. The possibilities are exciting. The question of integrating new technologies into the curriculum gave us the opportunity to review the larger questions of what we are trying to do when we teach calculus. This process has us reexamining the pedagogical methods we are using in core mathematics cores. Hopefully, the lesson we learn will spread up and down the mathematics curriculum.
• Abstract Algebra - I am of course interested in teaching my specialty. This has included the undergraduate and graduate sequences in algebra. It should lead to further advanced topics courses as well. As I teach these courses, I am exploring ways that various computer programs can be used to improve student understanding in these courses. The courseware is less available at this level, so the effort will involve more developmental work than with calculus.
• Technology in education - Since, as a mathematician, my focus is on how to use computers to use computers to teach mathematics rather than trying to teach computers, I find that there is a great need to think about when computing power can be used to illustrate concepts instead of providing a computing black box. There will also be a need to develop materials for the teaching assistants and new instructors so that the technology becomes a useful tool rather than just something else to cover. It is a challenge to find the right mix, being a professor who thinks routine drill, board races, group projects, and computer algebra systems all have an appropriate role in the classroom.

  My nonacademic professional life. As a Jesuit priest at a Jesuit university, I am also involved with a number of pastoral activities including:

• Saying Mass on Wednesday nights in Reinert hall, the dorm where I live as a campus minister.
• A regular rotation in the daily Masses at College Church
• The Sunday campus ministry and parish Masses at College Church
• Being available to help with student retreats and reconciliation services
• Whatever else comes up