I am currently two years into a math PhD program at a mid-tier state school. I read a lot of advice about getting a STEM PhD on the Internet, but a lot of it seems very inapplicable to mathematics. I don't have a lab or a PI, I don't need data, and my assistantship consists solely of teaching. None of the advice I read sounds anything like the experience I'm having here.

Compared to other STEM fields,

  • How common is it for math students to be supported by external funding (e.g. NSF GRFP) rather than a TAship? How common is it for math students to do internships in industry to gain practical experience? How common is it for math students to do outreach or volunteer work during graduate school?

  • Does a math department generally provide support and guidance to students when it comes to finding and applying for fellowships and internships? Is it a common attitude that it is the student's responsibility, if he wants those things, to undertake all steps of this process by himself?

  • Is it normal for math departments to disregard students' research interests in favor of mandatory coursework? For example, because all of my time is tied up in the 1st/2nd year courses and exams, I have been unable to do any research for the last two years, despite being very prepared and capable on my first day. (I published during undergrad and was chasing several promising ideas when I arrived.) When I asked to be allowed time to do research instead of taking classes, I was rudely shut down.

  • How often do mathematics departments get together in social or community events, such as departmental happy hours?

  • How is networking different in mathematics? Do professors generally have connections in industry or prominent members of their field, or is that a rare thing?

(Note. I'm sure it is evident from some of the above questions that I am feeling a little put off by my department. The point of this question is not to seek out sympathy or validation. The department just seems stubbornly uncooperative, and completely uninvested in my future, which is not what I was expecting. I want to know how common this is- whether it is unique to my department, or the nature of the discipline. But this is a peripheral point, the question I'm asking is about how these factors work in general, not just with me.)

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    ...I was rudely shut down. — Time to find a new department. Walk away. – JeffE Dec 18 '14 at 6:19
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    You ask many questions. Please ask one at a time. Also, please clarify that you are studying pure math or applied math? If pure math, why do you need internship? Internship to do what? – scaaahu Dec 18 '14 at 6:21
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    I know grad students in many different fields who did not have much time to devote to research in their first two years because of classwork. If that is what your program expects, it's fine. (If they're rude about telling you that, or you otherwise feel uncomfortable in the department, that's not fine, but that's a separate issue from what the program requirements are.) – BrenBarn Dec 18 '14 at 6:49
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    "...in favor of mandatory coursework". If the coursework is mandatory, well it is mandatory. Do you want to make an exception just for you? "..all of my time is tied up in the 1st/2nd year courses and exams". You did have summer vacations and weekends didn't you? A PHD is not a 9-5 job. I believe you could find a solid 3-5 months to work on your research in a two year period if you tried. There are people doing PHDs having families and jobs on the side. If they can do it, so can you. – Alexandros Dec 18 '14 at 9:43
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    I may not be qualified to answer this fully but with the limited knowledge I have and [your question][1] on math.stacksexchange , I really do not think that you are ready to start your own professional research yet. Instead, it sounds like a good idea to learn as much as you can in the graduate courses that are being offered, it will give you a better insight of mathematics as a whole and it may also influence your interests in the sub-fields. [1]: math.stackexchange.com/questions/1072910/… – Marko Karbevski Dec 19 '14 at 14:08

In brief, many aspects of math grad school are very different from CompSci, for example.

The vast majority of math grad students are funded by TA-ships, although some have NSF and other fellowships. The (large) size of typical math depts is tied to their role as service departments, teaching lower-division math, and this is what funds the large number of math grad students.

Internships in some applied math fields are desirable, but non-trivial to arrange.

Outreach and volunteer work are not strongly connected with these other questions.

It is not that math depts "disregard students' research interests", but that, almost universally in the U.S., further coursework beyond undergrad work is ... wise. If one has an unusually solid background, there are usually procedures by which to "test out" of requirements. On another hand, I am well aware that a certain number of grad students do feel that they're all ready to do research, and that coursework gets in the way. The "problem" is that "research" at a professional level in mathematics is not necessarily an immediate continuation of the sort of "research" typically done in undergrad projects. Another "problem" is that mathematics is an old subject, and there is a lot of very useful, helpful, enlightening stuff already known... and whose relevance to any given research project is very difficult to guess based on ignorance. I am absolutely not in favor of "oppressing" students by pointless busywork, but I am equally opposed to ignorance. Accurate perception of a given situation is difficult, and both students and faculty often have pre-existing biases... Again, testing-out of requirements ought to be a viable option, if one is well-prepared. If one doesn't see the relevance of the requirements, I'd tend to interpret this as reflecting a need to better understand the content of the required subjects. (Because they are relevant.)

In particular, depts to not "disregard students' research interests", except as not automatically exempting students from "requirements". Further, there are many hours in a day...

As to social events... it depends.

Networking? People know people. "Industrial connections" would exist only for very applied people, although connections to applied science research groups inside the university are common.

And/but none of what the question describes strikes me as unusually "uncooperative". In my observation, it is very common that math grad school is quite different from what people are expecting, exactly insofar as many students expect to immediately "start research" (perhaps parallel to the impression given about other STEM fields' programs), rather than having any required coursework at all. Also, there seems to be a not-uncommon disaffection with TA-ing, as though this were lowlier than having a fellowship of some sort or research assistantship... and is construed, again, as "obstructing research". But without all these TA-ships, many fewer math grad students would have any financial support at all. (There's little grunt-work available in mathematics that would compare to the low-level research-support work in some other STEM fields, which does (by tradition) get the student's name on a published paper, etc.)

So, yes, mathematics is somewhat different from other STEM fields. Further, the fact that a program is not what one presumed it would/should be is not at all a strong indicator that something's wrong with the program (although, of course, there are dubious programs). The greatest resentment I see is among students who believe that they're fully-fledged "researchers", and are offended to not be immediately treated as such... While this message can be imparted rudely, and perceived as "a rude shut-down", the many issues of professional competence are not easy for novices to judge.

(One more time: one can see about "testing out" of requirements...)

Edit: prompted by Brian Borcher's comment... Teaching is an important part of an academic mathematician's job! The question of "how much" is secondary. The TA experience is very important to get up to speed on teaching, and, in unhappy cases, to discover early on that one hasn't the taste for it, if that is so. (I had one PhD student who discovered this unhappy fact in his own case, so he did not take the academic route.) Part of what one should try to learn is that "teaching is not a burden"... !!! ... to say the least. :)

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    I'd add that for some reason many graduate students assume that teaching isn't really important, when in fact most of the new PhD's will end up teaching undergraduate courses for a living and very few will end up working in research. – Brian Borchers Dec 18 '14 at 16:11
  • @BrianBorchers, yes, indeed, that point deserved more emphasis in my answer! – paul garrett Dec 18 '14 at 16:22

While I think some of your questions point to genuine weaknesses in math departments (a lack of an internship pipeline, for example), a lot of them reflect a really unrealistic idea of what "cooperation" means. Your DGS, chair, etc. have met a lot of 1st/2nd year graduate students, and thus they know that very few of them are ready to do serious research, and in fact very few of them really know what that would mean. It wouldn't be looking after your interests to leave you to do research on your own, with no way of checking that you aren't just wasting your time. 1st year classes exist to cover basic material that any mathematician should know. If you know the material, it would have been reasonable to skip them (and usually departments will give students flexibility to do this), but as I say below, if they are taking up all your time, that gives the lie to your suggestion that you were already ready to do research.

How common is it for math students to be supported by external funding (e.g. NSF GRFP) rather than a TAship?

Rare, outside the very top schools.

How common is it for math students to do internships in industry to gain practical experience?

Rare in my experience in pure math departments.

How common is it for math students to do outreach or volunteer work during graduate school?

More common. I've known lots of people who taught (for example) in prisons as volunteers, or worked with K-12 students. Depends a lot on what's available.

Does a math department generally provide support and guidance to students when it comes to finding and applying for fellowships and internships?

For fellowships, they should but the reality is more mixed. In my experience, the problem is more that students aren't willing to go through the application process, so faculty have been burned a few too many times to be proactive about it. It might require a little initiative to get help with this.

Is it a common attitude that it is the student's responsibility, if he wants those things, to undertake all steps of this process by himself?

Yes. Graduate students are adults, and responsible for themselves. It might happen that someone in the department is looking for graduate students for an opportunity like this, but at the end of the day, it is up to you.

Is it normal for math departments to disregard students' research interests in favor of mandatory coursework?

This is pretty insulting: obviously the department requires the coursework because they believe it is in the students' interests. Of course, it's hard to have a system that fits everyone, but very few students could be successful in grad school in math without taking a couple of years of classes (again, with the possible exception of the best schools).

For example, because all of my time is tied up in the 1st/2nd year courses and exams, I have been unable to do any research for the last two years, despite being very prepared and capable on my first day. (I published during undergrad and was chasing several promising ideas when I arrived.)

If you were very prepared and capable, then your first year classes should have been easy, and you should have had plenty of time. If they were hard enough to take up all your time, you didn't know the material.

When I asked to be allowed time to do research instead of taking classes, I was rudely shut down.

I can't comment on whether it was rude or not, but I can't say it was a surprise that this approach did not work. Since you haven't mentioned a research advisor, I'm going to assume you don't have one. Very few students would be capable of doing this, and I have trouble imagining that your undergrad publications were so strong and independent that the department should have had faith that you were one of them.

How often do mathematics departments get together in social or community events, such as departmental happy hours?

Depends. Departmental happy hours aren't a strong tradition, but department teas are, and usually departments have a department-wide social event like a picnic or holiday party every semester or so.

How is networking different in mathematics? Do professors generally have connections in industry or prominent members of their field, or is that a rare thing?

Rarely in industry, usually with prominent members of their field, but this depends a lot on the definition of "prominent" and "field."

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    It is impossible to emphasize too strongly: "Graduate students are adults, and responsible for themselves." It is often challenging to shift from an undergraduate mindset to a professional one, but the sooner the shift occurs, the better chances the student has, in my opinion. – Oswald Veblen Dec 19 '14 at 15:13
  • The thing that I don't like is that it seems like I'm only an adult when it's convenient for them. If I want to manage my own learning, which I've been waiting to do for a long time, then I'm inexperienced and don't know what's good for me. If I request help securing funding, I'm an adult who should have known about this 10 years ago. Really it seems like the whole "be an adult" thing is not so much a criticism of my behavior as it is a way of displacing blame whenever they don't feel like living up to their part of the bargain. – Hurwitz Carlton Dec 20 '14 at 5:46
  • I also want to say that though I haven't picked an advisor, it is between three professors, with whom I have been working on research problems during my sporadic free time. My research was published in a top journal as an undergraduate, and I was the sole author. I tested out of coursework in my field. The reason I'm having trouble with other courses is that I'm having a troubled time dealing with the growing resentment I feel from being overbearingly micromanaged. The material is not hard, but I would be much, much better off teaching it to myself. I know how I work. – Hurwitz Carlton Dec 20 '14 at 6:04
  • Anyway, I want to thank you for posting this answer, and addressing all of my questions. (I'm sure my response sounds bratty, because I'm venting frustration, but I don't mean to direct it at you.) Again thank you genuinely for the response. – Hurwitz Carlton Dec 20 '14 at 6:08

I'm going to focus on the third question: "Is it normal for math departments to disregard students' research interests in favor of mandatory coursework?"

In comparison with PhD programs in many other fields, PhD programs in mathematics in the US typically have quite a bit of required coursework and often include preliminary/qualifying exams that cover a very broad range of topics. In my experience, graduate students working in other STEM disciplines typically have fewer required courses to take. Thus the answer to your question is "Yes, this is quite common."

For example, as a graduate student I took required courses in abstract algebra, real and complex analysis, functional analysis, and topology, even though my dissertation was in computational optimization.

It's traditionally felt that mathematics PhD's should have a broad background in mathematics so that they can easily teach undergraduate courses in almost any area of mathematics. The required coursework helps to develop this breadth.

Keep in mind that the academic job market for PhD's in mathematics in the US is very different from the job market for PhD's in other STEM disciplines. Most PhD's in mathematics will end up working in teaching positions at community colleges and regional comprehensive 4 year colleges rather than in research oriented positions at universities. Having a broad background in mathematics (and experience as a TA) is helpful preparation for teaching oriented positions.

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    In my mind, it's not only that the students may need to teach in various areas. Part of holding a PhD in mathematics is having a broad expertise in the field. So, for example, even someone who will only study finite combinatorics will often need to pass exams in real analysis. Having a PhD in math means something both in terms of depth of knowledge (in the dissertation) and breadth of knowledge (in coursework and exams). – Oswald Veblen Dec 18 '14 at 18:38
  • My goals are purely research, whether be in academia, government, or industry. Do you think that leaves me out to dry, if my field is mathematics? – Hurwitz Carlton Dec 19 '14 at 7:22
  • I should clarify that I don't mind teaching. In fact, I like it and think it is very important. But I would consider teaching secondary to my research goals, to be realized as a peripheral benefit if I take the academia route. – Hurwitz Carlton Dec 19 '14 at 7:24

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