# Teaching advanced courses in a specialized area of math [closed]

Today mathematics (and probably also other sciences) became very specialized: every researcher works in their own narrow field which usually hardly interacts with fields of other colleagues, even on the same department.

At the same time every researcher is often expected to accept PhD students and introduce them into his/her field. Some mathematical fields (e.g. algebraic geometry, algebraic number theory) require significant background before starting the research, and hence teaching quite a few advanced courses.

How this problem of regular teaching of many advanced courses on a certain subject is resolved on your math department? Who teaches all of them? Does there always exist a group in your area on your department so that the teaching load can be shared? How it is decided who teaches what? (Please indicate your country.)

• Likely math isn't essential to the question. Pretty much every field requires deep dives into narrow canyons. – Buffy Sep 26 '19 at 15:01
• @Buffy I worked in pure math and also very applied math. The amount of prerequisites to really start doing research was vastly different. – J. Fabian Meier Sep 26 '19 at 18:29
• I know that I could just compare two fields. But when I worked in applied discrete optimization, it was relatively easy to understand problems and methods from other people. It was even possible to explain my research to my girlfriend who did Algebraic Topology. In pure math, I was often lost in research seminars when the topic was about another subfield. – J. Fabian Meier Sep 26 '19 at 19:13
• @J.FabianMeier: Here's a different way of looking at it. Let's define a subsubfield to be a group of people who mutually all understand each other's research. Then, in applied mathematics, subsubfields are large, because there is quite a bit of funding to look into any particular topic, and it's possible to have a research seminar series where everyone is in your subsubfield (and hence, by definition, understand each other's work). In comparison, pure math is poorly funded, so subsubfields are small, so any particular seminar series is going to have speakers from many subsubfields. – Alexander Woo Sep 26 '19 at 19:31
• @AlexanderWoo It is not just that. If you compare the number of years you need to study until you hit the cutting edge of research, you find huge differences. – J. Fabian Meier Sep 26 '19 at 19:36

As a combinatorial algebraic geometer in a small, not highly ranked department who currently has a PhD advisee, I think I can say something about our experience.

We do not get to regularly schedule advanced courses. With 14 grad students total, there aren't enough students to make teaching algebraic geometry viable. We offer a 2 year sequence in general algebra, commutative algebra, and algebraic geometry on an alternate year basis, organizing the courses so that it is possible to start midway at the commutative algebra entry point. When we have a relatively capable group of students, we just might be able to get to the definition of a scheme in the last 2 or 3 weeks of the last semester.

If we happen to have 2 or 3 graduate students at the same time who are interested in a particular topic (for example algebraic topology, or more advanced algebraic geometry, or algebraic number theory), we may organize a course for them, but faculty usually have to do this "for free" - in the sense that whomever is doing this teaching is teaching this as an extra course in addition to what he or she normally teaches.

My advisee has answered an algebraic geometry question well enough to write a dissertation. He has not taken an algebraic geometry course. How can he do this? He has learned just enough of the specific bits of algebraic geometry (really just commutative algebra) he needs to tackle this specific question. Fortunately my area attracts enough interest from combinatorialists and representation theorists that there are some reasonably accessible books and papers, but he has ended up learning what he needs to know mostly from me in our weekly meetings. (Yes, I am disputing the claim that one needs a lot of background before starting research, even in a fairly esoteric field.)

Does he understand why the question he is working on is of interest to me and to other people I work with? Not really. Will he be able to find related questions and answer them in the future without my guidance? I am skeptical he knows enough mathematics around his current problem to be able to do so, but maybe. He is applying for teaching-oriented positions where the point of doing research is to provide research opportunities for (undergraduate) students rather than to advance knowledge, so maybe his research experience is sufficient.

• Let me suggest that you help your advisee build a circle of collaboration with you and others with similar interests who you know. Even in a small department it is possible, today, to have such a synergistic group to share ideas. It would be of use to everyone involved, but especially to your student and his students. – Buffy Sep 26 '19 at 20:46
• In similar contexts, I think this happens very often. It's undesirable, but inescapable to some degree, hence, necessary. In a not-so-conducive environment it does take a lot of energy to promote ideas consistently, as I have noticed in various scenarios. And, indeed, other people are reasonably-enough looking after their own agendas, and perhaps do not have unlimited energy. Small departments are tricky in many ways. – paul garrett Sep 26 '19 at 21:17
• That's a great answer, with a lot of local "color". – guest Sep 26 '19 at 21:29
• I personally think tough, good courses are the way to go, even in advanced fields. Universities generally don't have the funding for such low course size work. But let's not kid ourselves about the gap. In contrast, you can look at how NASA, the military, and the NFL do hands on instruction even for advanced performers. But of course this is very expensive. – guest Sep 26 '19 at 21:29

Perhaps there some of the curricular aspects are more manifest in mathematics than in some other subjects, namely, that "old things don't go away", for one. For another, the "going ever deeper into ever narrower topics" is not a good description of modern mathematics, considering that interconnections between ideas, especially seemingly disparate ideas, is a big part of modern mathematics. Also, there is a general tradition in mathematics in the U.S. that people have a bit broader background than just the minimum to scrape by in "their specialty", even if they are deliberately aiming for a narrow specialty.

With that in mind: also, basic courses in algebraic geometry or algebraic number theory or functional analysis or modular forms or Lie theory or... are not really "advanced" except in comparison to undergrad courses, after all. For example, my own PhD students would benefit from taking all these, and more.

And, yes, at my large-ish state university, there are rough groups of people interested to varying degrees in all these courses, so there's no lack of enthusiasm to teach them, although precise organization is always a minor issue. Many of us think of teaching such courses as quite a lot of fun, because these things verge on "real math", even if they are still introductory and worked-over.

But, yes, in a smaller department, and perhaps with less research emphasis, and perhaps a smaller PhD student population, the interest groups for many things might not quite reach a critical mass for such courses to be reliably viable.

• While I don't disagree with the things you say, this seems more like a rebuttal of my comment than an answer to the question of the OP. "How does it work?" I suspect you have insight into that. Your profile suggests you are a denizen of the esoteric deep. – Buffy Sep 26 '19 at 18:50
• @Buffy, not intended necessarily as a rebuttal to your comment, but, yes, incidentally mildly so, while also mildly disagreeing with some of the premises of the question itself... I did attempt to answer the "how does it work?" or at least "how can it work?" – paul garrett Sep 26 '19 at 18:52
• @Buffy "denizen of the esoteric deep" :) – paul garrett Sep 26 '19 at 21:22
• You can put it on a T-shirt if you like. – Buffy Sep 26 '19 at 21:33

In France, that's what the master's degree is for. More precisely, the second year of master (colloquially known as M2) with the specialization as research master degree, as opposed to teaching master degree or professional master degree.

Advanced courses are taught during the M2. There is a huge gap between the level of the licence (bachelor) and the first year of master (M1), and another huge gap between M1 and M2. Students are expected to work a lot on their own. Moreover, in every math M2 program, the student is expect to do a 3-4 months "internship" with a professor and write a "memoir" at the end, a mini PhD thesis. This is very often where you get to learn the very advanced, very specialized material necessary to start doing PhD level research.

I should now state that there is a very clear difference in organization between Paris and the rest of France. France is extremely centralized, and this shows in math too. There is very large supply of M2 students in Paris, with people coming from Grandes Écoles or abroad just for the M2; meanwhile, outside Paris, M2 programs usually barely manage to get by with 5-10 students a year. I've experienced both (as student and faculty) so I can comment on what I know.

How this problem of regular teaching of many advanced courses on a certain subject is resolved on your math department?

In Paris, there is a wide variety of M2 courses on most subjects. There are enough students and faculty for this.

Outside Paris, not all courses can be taught every year, as there aren't enough students and faculty: some classes would be empty, some would lack an instructor (thanks to massification of higher education, we have to teach a shit-ton of classes to 1st year undergrads). A usual deal is to do thematic years, for example "this year we'll do a number theory program and a stochastic equations program. Next year, algebraic topology and dynamics." If a student wants to study something not offered, they unfortunately have to move.

This difference is also magnified by the rarity of PhD stipends. What good is giving someone a research master's degree in math, a degree that only prepares you to become a professional mathematician, to someone who has no chance of getting a PhD stipend afterwards? As you can guess, some universities are richer than others and can offer more stipends.

Who teaches all of them?

Faculty…? Who else? I don't understand the question.

Does there always exist a group in your area on your department so that the teaching load can be shared?

In Paris, yes. Outside, no. As I said, we have to teach a ton of classes to undergrads. And somehow, no one in the government was able to divine that babies born in 2000 would be college students today, so they haven't had the foresight to increase budgets for faculty jobs. So we are stuck with a dilemma: enrolling students in undergrad is something that we have to do by law, the number of students increases every year, and the budget stays constant. Guess what part of higher education is slashed first?

How it is decided who teaches what?

There cannot be a single answer. Every department in France has statutes and bylaws that can be (democratically!) changed, and this includes how teaching loads are divided. Still, every university I've been to has had this model: the department director delegates the power to either a committee or even a single person depending on the size of the department and customs. This committee/person deliberates on what courses can be offered and who teaches what. It's often (but not always) taking into account the wishes of every member of the department. Large department (= Paris…) have even more complex procedures.

PS: I hope this does not come off as an answer from a Parisian snob. The math departments in hard science-intensive universities in Paris are about 2-3 times as large as the largest math department outside Paris; moreover, it's only a metro ride away to the other Parisian departments, who can thus coordinate among themselves to have a unified program with much wider breadth. I have many colleagues outside Paris who despair at the state of their M2 program (and this won't be helped by the current government's policies of creating monster universities on the one hand and defunding smaller ones on the other hand… if you know how to solve this problem, please tell us).

PPS: This answer is mainly about pure math. The situation is less bleak in applied math. ("What good is pure math for anyway? Pure mathematicians can't get a contract with Thales or Safran to co-supervise a PhD student who will help us sell bombs to Saudi Arabia." --some government official, probably.) Some smaller math departments are slowly transitioning to applied math only.

In chemistry, classwork during the Ph.D. is pretty minimal and the emphasis is on doing lab work. There is a vast amount of even foundational stuff that is not well covered (especially in instrumental analysis). I believe the trend has been towards less courses over time. This is generally perceived by students as a plus (not by me, see it as more technician work, less learning), and also by PIs, who are incented to run large groups of workers, rather than training scholars.

My impression is the same is true in biology. [Lots of funding for grunt work...gotta cure cancer, right?] Less so in physics, where there are tough qualifier exams.