I was reading recently online Peter May's complaints (I'm a fan, you can tell, I'm sure) about teaching the third quarter of the graduate algebra sequence at the University of Chicago. This course focuses on homological algebra and attempts to be as up-to-date as possible. May's conundrum stems from the fact that homological algebra is inexorably tied to algebraic topology and as a result, it's difficult to separate the 2 out in the course completely. May questions whether or not this is in fact a good idea; however, since this an algebra course and not a topology course, he feels compelled to work hard to do this.
That being said, he raises a very good pedagogical problem in the teaching of mathematics, particularly at the graduate level where the better schools are trying to prepare students to enter research as quickly as possible. Mathematics is now a very holistic, intertwined discipline: Algebra increasingly permeates virtually all of mathematics, the study of manifolds now requires very sophisticated analytic tools from differential equations and functional analysis, probability theory now partakes of a considerable amount of harmonic analysis, mathematical physics is now a major player in the construction of new mathematical structures-I could go on and on, but you get the idea.
So here's the question: Is the old model of keeping the subdisciplines of mathematics separate in coursework for the sake of focus obsolete? I know a lot of mathematicians in recent decades have begun to draw from various disciplines in constructing the first year graduate sequences of most universities; Columbia is one local example. The question is really are they going far enough? The problem of course is that when you begin weakening those artificial barriers, you run the risk of them collapsing altogether and you ending up with a hodgepodge of theory and methods that seems to have no focus.
So anyone want to comment on what the solution here might be from their own experiences as both teachers and students? How far should courses go in being interrelated? And does this lead to better prepared graduate students for the research level?
Best Answer
Of course you should show students, taking into account their backgrounds, that the material they are learning in one course is relevant elsewhere. It makes it clearer to the students that topics they are studying have wide usefulness. At the same time, if you know the students don't have a background to appreciate the technicalities coming from other disciplines (not everyone in algebra has had algebraic topology), then you may have to restrict yourself only to making some broad general remarks, although maybe one or two special worked examples from the other disciplines would be accessible without a lot of machinery.
When I discussed characters in an algebra course, I explained a little about Fourier series both for context (otherwise the concept can seem rather far-out) and so they'd see that the otherwise idiosyncratic theorems on characters are related to properties of Fourier series.
I don't think such discussions in a first-year course are going to make the students better researchers, but it will make them better appreciate what they are supposed to be learning.