The interrelationship problem of modern mathematics – How to deal with it in first year graduate courses?

Question

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?

Answer 1

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.

Answer 2

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?

Answer 3

I've almost uniformly studied the homological algebraic aspects before I got around to studying the corresponding results from algebraic topology. It did get somewhat artificial at points - specifically triangulated categories make a lot more sense once you've seen Serre fibrations than before you do.

I felt quite well motivated by the approaches I encountered though; with the study of Ext and Tor to divine interesting ring properties taking the forefront in homological algebra, with a side dish of approximating modules by things that are free everywhere that matters, but sacrifice degree concentration to achieve it.

My personal feeling is that it probably depends to a large extent on whether whoever is teaching the material wants to teach homological algebra or algebraic topology: if you're happier thinking about topology, then homological algebra will feel desolate and artificial almost no matter what you do about it; while if you are genuinely interested in homological algebra on its own, it's much easier to sprinkle in the off-ramps as you go, pointing out where certain concepts have roots outside the current area, and how to get more information about the roots.

Answer 4

I am only a first year graduate student, but I am very interested in mathematics education. My own approach to teaching is very much problem based: give students interesting problems which lead to the development of the concepts you want them to have. Even if they can't come up with all of the needed concepts on their own, if you give it to them after they have wrestled with a problem they will be much more likely to be able to apply the concept in novel situations in the future. Why couldn't this approach be carried through in a math grad situation? Design a sequence of problems, varying in difficulty, which in total cover need material from most of the "first year curriculum".

Before writing this off as a crazy idea, I would like to point out Cornell's vet school. They use exactly the model given above: Every week or two there is a new case. In each of your classes (anatomy, pharmacology, radiology, ...etc) you cover general information which is pertinent to the case of the week, but it is up to you and your team to do research, come up with a diagnosis and a method of treatment. So all of the classes you take are integrated together in the context of solving some real problems. Cornell is turning out some amazing vets. Why couldn't the same model work for mathematics?

Answer 5

This is an old question, but I only just stumbled across it.

I agree that it's important to convey to graduate students the inter-connectedness of mathematics. But I don't think that the way to do it is to dissolve the traditional boundaries between graduate courses.

The best connections between different fields of mathematics arise when deep calls to deep; i.e., when a deep result in one field connects with a deep result in another field. So a prerequisite for such connections is a deep development of individual fields. And the only way to develop an appreciation for depth is to stay focused on one thing for a period of time and develop some degree of mastery over it. Too much jumping around without any deep dives runs the risk of superficiality.

Of course, this doesn't mean that graduate courses should be siloed. When an opportunity to mention a connection with some other field arises, by all means seize it. Another MO question, Your favorite surprising connections in mathematics, provides a long list of potential examples. One example from that list that I like, and which is related to topology, is the chromatic number of the Kneser graph, whose proof is simple enough to cover in full in one lecture, if students already know the Borsuk–Ulam theorem. But in general, it is not necessary for the instructor to take a lot of lecture time, or prove all stated claims; students will derive considerable benefit just from knowing that the connection exists.

Homological algebra—the example that the OP began with—is in my opinion a bit of an outlier, in the sense that it's a topic where it's all too easy to spend an entire semester just developing machinery, without giving students any sense that the subject is (arguably) more interesting because of its ability to solve problems in other disciplines than as a subject of interest in its own right. For this particular course, it may be worth experimenting with designing it "backwards"—picking some particular applications and developing the machinery needed for those applications, even if doing so forces one to skip some traditional topics. Algebraic topology is of course the most obvious source for applications, but if the students don't have much background in topology then there are also beautiful applications in commutative algebra, such as the Auslander–Buchsbaum theorem.