TITLE: The Interrelationship Problem Of Modern Mathematics- How To Deal With it In First Year Graduate Courses?
QUESTION [27 upvotes]: 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?

REPLY [23 votes]: I question whether mathematics is really as holistic and intertwined as some people are making it out to be.
Certainly there are a few freaks of nature out there who can understand 20% of the mathematics out there and incorporate ideas from 10 different subfields into their work.  A larger group of us are capable of getting the big picture though maybe not all the specifics of 4 or 5 different subfields, at least to the extent that we know when to reach out to an expert.  Many graduate students, and most of them once one leaves the world of the top ten or twenty departments, are just capable of learning one subfield well enough to write a dissertation narrowly focused on one problem in that subfield, ignoring all the wider connections if indeed there are any.  Most published papers are written by people who have never done serious work outside a single narrow subfield in their entire career, even if the same is not true for the best papers.
A professor or a department may choose to aim its education at the future Fields Medalist (or, somewhat more broadly, the future NSF-or-equivalent-research-grant-receipients), but is this really fair for the other 19 students in the room?