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Fresh from undergrad, I will be starting a math PhD program this upcoming fall in the United States. After several conversations with math professors and current PhD students, I noticed that several people advised having a sense of "urgency" right from the very start of your PhD program. Most recommended working to complete preliminary exams as soon as possible, but there was some variance in their opinions about when to jump into your thesis. One professor recommended "courting" potential thesis advisers by doing research projects with a professor or two during the first couple summers.

Do you agree with this call for urgency? If so, how do your recommend attacking those first 2-3 years of a (pure) math PhD program? Is it ever too soon to decide on your thesis topic? Recall that I do not have a master's, although I have taken three graduate-level algebra courses.

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    @JeffE: In the year 2016 teaching needs to appear on your list somewhere. This is now very important for graduate students. Two days ago I attended a meeting to apportion government grant money as summer funding to students. After careful deliberation we decided to tell a student that he was not getting the summer funding precisely because his teaching had been problematically poor, though his research has been very good.
    – Pete L. Clark
    Apr 2, 2016 at 21:28
  • Hi Professor Clark, does that mean being a TA / tutor / grader during one's undergraduate / master's degree studies is pretty critical and that PhD admissions committees will look for this experience? As critical as having top grades in coursework + a good research project / thesis? Thanks @PeteL.Clark
    – User001
    Apr 3, 2016 at 4:22
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    (After correction by @PeteL.Clark) The five most important things to focus on, in decreasing order (but don't ignore any of them) — (1) Research. (2) Teaching. (3) Research. (4) Administrative hurdles, like classes and exams. (5) Research.
    – JeffE
    Apr 3, 2016 at 12:54
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    @Chris: This sounds like a good separate question. A quick preview: "I have only been able to get 3 quarters of TAing in 3 years." I see nothing to complain about there. For math grad student teaching: (i) you should do some; (ii) you should do all of it adequately and make best efforts to have at least one course go very well; (iii) subject to (i) and (ii) you want to teach less, not more. I taught (not TA'd) 5 courses in 5 years as a graduate student and had 6 semesters without teaching. If I got to pick my own teaching load given what I know now, I would pick 3 courses instead.
    – Pete L. Clark
    Apr 3, 2016 at 21:40
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    @User001: No, my comment was entirely about people who are already in a PhD program, not about those who are applying for one. In many institutions you need to be a graduate student to do any TAing/teaching whatsoever, so that's that. Admissions committees do not look negatively on a lack of teaching.
    – Pete L. Clark
    Apr 3, 2016 at 22:08

4 Answers 4

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I'm a third year, and I have found it immensely helpful to get started with research as quickly as possible.

The thing is, a lot of times you get stuck in mathematical research, especially when you're first starting out. Also, you will have a harder time finding a concrete problem then you'd think. Additionally, you could use your research to guide which classes you take (at least after quals). Lastly, making progress is hard and takes time. This means that the best way to do research is to always have a research problem.

It may not be your main focus, your main focus may be your exams, but if over your first two years you've spent random times thinking about a specific problem (and have geared your studies towards a problem), you can begin to make a lot of headway. So I highly suggest you start the research process. Talk to some advisers. Find something that interests you. Read some papers in your free time. Use this to find an interesting problem. Talk about this with someone who researches the area and now you have an adviser. And just start thinking about it. This is all pretty leisurely and can be done in your "free time" between classes, after classes, etc. Don't worry about the time consuming parts (actually writing down solid proofs), just get your mind churning.

When exams are done, you'll be surprised how ready you are to write what you've come up with. Even if you haven't solved much, you have likely narrowed down a clear path of where you want to go and what you want to prove (and have an intuition as to why you think it's correct). Write down that path as a "homework problem" to yourself, and now you're full-time research.

(Some people will tell you to wait until you can understand your project. Honestly, I didn't understand what I was doing (stochastic dynamics) for about half a year. After finally taking the classes on subjects related to the field, I can say that classes wouldn't've helped much. The only way to get into a research project is to just start doing it, and keep trying different books/resources until you understand all the background. Classes help, but less than you'd think.)

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I think the relevant sense of urgency would reasonably be about engagement with serious mathematics, not necessarily keeping a pretense that one has "started research" at a quasi-expert level. Nor do I personally see urgency in "passing exams", since that is a typically very-limited enterprise, but might take long enough that one's original interests, enthusiasm, and motivations are forgotten, which would be a bad thing.

And, yes, standard coursework tends to suffer from so much inertia that it is not reliably helpful. But that is not to say that there's little to learn. In principle, sure, one can learn everything needed from papers, books, etc., whether physical or electronic. However, there is so much of this material that some guidance is good. Preferably expert guidance, rather than novices-leading-the-novice.

In particular, in terms of "choosing research problems", while it's good to follow up on one's enthusiasms, one should not be surprised to discover that, very likely, an enthusiasm based on just-a-little information inexpertly chosen is ... naive. Of course! Seemingly natural research problems are all too easily accidentally intractable, or perceived as routine exercises by experts, or of no interest to experts, etc. If one is independently wealthy and just doing math for personal satisfaction, none of this matters. However, if a PhD is in part warm-up to making a living as professional mathematician, the expert-standards do matter, however artifactual they may be.

(The issue of "understanding" one's "project" is ambiguous! One may doubt on general principles that one understands fully, indeed, or suspect that one understands to-some-extent naively. That's entirely reasonable. I would claim that much of the work done to finish a PhD in mathematics these days amounts to getting up to speed on techniques and viewpoints that experts might declare "standard" or "routine"... since, after all, some facility with such methods does sometimes enable one to do things that would seem amazing or unbelievable from a more low-tech viewpoint. After all, that's most of the point of sophisticated methods and ideas.)

So: the urgency is to interact with experts as early as possible, yes.

But, again, it is not accurate to say that this excludes coursework, nor includes it, either. It does not exclude or include following one's own curiosity. It is a separate thread.

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    I forgot to make the usual point that some of the purpose of "coursework" is to become acquainted with the style and attitudes of the faculty who teach those courses, whom otherwise one might not have interacted-with. I do claim that the literal content is not purely objective, e.g., its significance (beyond being traditional, iconic, whatever). Further, the tone of voice and visible affect the instructor has in reaction to given content conveys important nuances, I think... assuming that the instructor is expressive, rather than impassive.
    – paul garrett
    Apr 2, 2016 at 21:43
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I'll first state I'm only a first year student in a math Ph.D. program so it's up to you to decide how serious you want to take my words.

I'd say yes to "sense of urgency" you describe in your question. Almost all professors I meet do tell me the first thing to do is to finish the exam requirements(preliminary/comprehensive, qualify) ASAP. They see passing exams as a "proof" of a student can do Ph.D. and will only take that student seriously after the requirement is finished. In addition, it's hard for Professors to give you any serious research work if they don't even know whether you will stay in the program or not.

That being said, it's not saying that you should never think about your research, another common advice I got is attending many seminars to explore my interest and determine the direction of my future research works.

It looks like you somehow already decide the direction of future works. Do you have a few potential advisors in mind? They are the people that can answer your "What to do in my first several years?" question accurately. Besides some common general advices and department's policy the answer of that question depends a lot on individual's condition and research field.

I notice you mention you've taken three graduate-level algebra courses before. Have you considered taking qualifying exam in algebra in the very beginning of your graduate study if that is possible (Of course this depends a lot on departments' policy)?

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    There is one professor with whom I am pretty sure I could work. And I have four attempts to pass two general exams, so I plan on attempting at least the algebra exam at the end of my first year. Because I have not taken any of the algebra sequence at that school, I don't want to risk getting a strike against me as soon as I get there. I would like to finish the general exams by the end of my first year, though.
    – Future Algebraist
    Apr 2, 2016 at 20:57
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A PhD program may prepare you for research. Richard Hamming in his talk “You and your research” asked the three following questions:

  1. What are the most important problems in your field?
  2. What are you working on?
  3. Why aren’t they the same?

They can guide you well, whenever you take a step in your first days of a PhD.

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    Might be more persuasive if the second question were completed in the second slot, etc., ... Also, some commentary on the literal-ness or not of these quips? E.g., are you telling kids to declare their PhD project to be P=NP? The Riemann Hypothesis? ... and then try to come up with an idea? Hopefully not. But then how to interpret these catchy quips (which do have some content, yes, but not their seeming literal assertion, I think)?
    – paul garrett
    Apr 3, 2016 at 1:05
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    This advice could be dangerous to a beginning math PhD student who is naive enough to think it might apply to them. I've read (more than once) Hamming's talk and find it interesting and useful, but his talk is pitched at research scientists, not beginning math PhD students. A beginning math PhD student doesn't yet have a field (other than mathematics, but since Hamming did not himself work on "the most important problems in mathematics" that is surely not what he meant), and what they know of important problems is only through relatively-popular culture, rumor and accident....
    – Pete L. Clark
    Apr 4, 2016 at 1:55
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    ...A math PhD student needs to spend at least a year trying to get a clue as to what the objects that they will spend later years of their life studying might possibly be. As an example, my first semester in graduate school I decided to attend a course on abelian varieties. On the first day the instructor reviewed the cohomology of complex tori, and upon prompting a more senior student immediately answered (in a strangely bored voice) "It's the exterior algebra of the lattice." Alarmed that I was so completely lost, I dropped the course. Later I wrote my thesis on...abelian varieties.
    – Pete L. Clark
    Apr 4, 2016 at 2:00

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