Welcome to Karl-Dieter Crisman's official webpage.
I am Assistant Professor of Mathematics at Gordon College, on Boston's North Shore. I teach a variety of math courses running the gamut from core courses to senior-level ones. I'm fortunate to have support from students through administration to:
- Be allowed to develop several truly interesting courses which are also inspiring to our students, such as a non-major course in the mathematics of voting and choice and a serious number theory course with a broad point of view (calculus to geometry).
- Do research in, teach, and guide students in research in the mathematics of Voting and Choice.
- Conduct independent studies in areas such as Representation Theory and Lie Groups with talented students heading to PhD programs.
- Attempt needed innovations in core courses, including continued development of a truly conceptual one-semester calculus course, and a service-learning component in mainstream calculus.
- Work on resources and code for the open source math program Sage.
I received my PhD in Mathematics from the University of Chicago; my thesis, "Chow Groups of Zero-Cycles Relative to Hyperplane Arrangements", was completed under the direction of Spencer Bloch.
Current research projects include:
- Work on mathematics as applied to voting. My two most recent projects are the following:
- Trying to characterize the set of profiles which arise from nonparametric statistical tests. This extends work of Saari and Bargagliotti. My student Sarah Berube has proved a nice consistency result, and I was able to prove some first combinatorial criteria for how a 'data set' gives rise to a specific profile decomposition (in the sense of Saari). This will appear in Mathematical Social Sciences; an earlier version is here, and here is an invited talk I gave in Irvine about this.
- Orrison, along with his students, has extended much of Saari's decomposition work into a more explicitly algebraic framework, using irreducible subspaces of the profile space under the action of the symmetric group to characterize various voting rules. My work (preprint here) extends this to the case of 'scoring' social welfare functions (Conitzer's SRSFs, and one version of Zwicker's generalized scoring rules). Here is a recent talk I gave about this. It turns out that the representation theory of the symmetry group of the permutahedron is the right tool, and yields a new one-parameter family between the Borda Count and Kemeny Rule (as social welfare functions).
- Math applied to music theory, particularly reinterpreting the geometric work of Callender-Quinn-Tymoczko in algebraic terms.
- Continued work on hyperplane arrangement invariants.
More pedagogical work includes:
- Service-learning work by students in several different calculus classes. I helped organize a special session in this at the 2011 Joint Mathematics Meetings. All the papers are archived on my site with the gracious permission of the speakers; take a look!
- This "Notebook" for courses of the Bridge or Transitions variety, an article about which is to appear in PRIMUS.
- A brief article introducing the symmetries of the permutahedron to abstract algebra teachers, to appear in CMJ.
- A draft of a number theory text using Sage.
- Many other talks about Sage in an educational context; the Sage worksheet for the most recent one (at the Clay Math Institute) is here (you'll need to create a free account to actually try out the examples).
- For on-campus viewers (only), a streamed video of a talk I gave in Faculty Forum as an (entertaining) introduction to the Math of Voting and why we might care about it.