Lots of teaching things going on right now! David and I were tasked (with extremely short notice) with developing an online abstract algebra course (entitled ‘algebraic structures’) for second-year undergraduates, and have been hashing out a good direction for the course to take.
One of the main things we realized (in a few hours of back and forth) is that the usual first course in abstract algebra probably isn’t really locally appropriate. This is largely because we can’t be assured that the later maths classes — which hook abstract algebra into a wide variety of different contexts — will ever be available to the students. The other big idea we came up with is that it would be great if we got students to the point where they could look at a mathematical problem and find the influence of algebra, and (in a perfect world) craft solutions to the problem appropriately.
This second big idea is something that really drives mathematics research. A major component of early homology theory was noticing that an algebraic structure called a Hopf algebra exists, and that homology theories are basically Hopf algebras. In my own work with Catalan-type objects, a recurring theme is that people study these things as combinatorial objects, without noticing that there’s a monoid structure there, and that we can actually understand these objects better by looking at the representation theory of the relevant monoid. Greg Musiker introduced me some years ago to a really nice paper developing voting theory using representation theory of the symmetric group: a full ranking of candidates is just a permutation, so we can use permutation groups to better understand different voting systems.
Finally, this ties into the seminar series I’m giving this semester, following some notes of Persi Diaconis. The notes use group representation theory to solve interesting problems in probability and statistics. Basically, permutations and other groups rear their heads all over the place in probability; understand the representation theory of these groups and you get a much better understanding of the probabilities.
So the upshot is I’m cranking out some new notes on basic abstract algebra. The hope will be to change emphasis over the course of the semester from the standard run of an abstract algebra course in order to prioritize some basic representation theory and seeing a couple nice `real’ applications. The usual application you see in an intro course is showing that certain positions in the ’15-game’ are unreachable; it would be nice to hit something a little more interesting!