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I would like to apply for graduate school to become a mathematics professor.

However, given that the opportunity cost of grad school is high and that I already have a stable, good paying job, I would like to know whether I have the talent and ability to be a math professor before I start. This is obviously a very difficult question to answer, but I'll explain my situation.

I believe I have above-average math talent, although I wouldn't consider myself a genius and I've had people in my classes smarter than me. However, I've gotten high marks in nearly all math courses and I've written a honors thesis which has also received very high marks. I've also worked as a research assistant. Basically, I think I've done pretty well and perform competently in all the mathematical challenges that has come my way so far.

But being a math professor requires original ideas and lots of publishing. I haven't had any significant original ideas, but it's probably true that most people at my level of education also haven't (correct me if I'm wrong here). My main concern is how do I know if I would be able to generate enough original ideas to keep publishing and maintain a successful academic career? Unlike working in mathematical modeling in the private sector, where one cannot really get "stuck" in the same way, it seems like a risk to be a professor since it's really hard to guarantee that you'll always be able to produce new research.

Do any mathematicians working in academia have any comments about how one knows if they'll be able to continually generate new ideas to produce publishable research?

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I don't think anyone knows, we just hope we can. –  StrongBad Feb 7 at 13:23
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"It's really hard to guarantee that you'll always be able to produce new research." The more serious risk is that you won't get a job. Be warned that there are more capable Ph.D. graduates than faculty positions to fill. –  Anonymous Feb 7 at 13:47
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Mathematics isn't something you are gifted with; it's a practiced skill. The reason you have above-average talent is because you do your homework, and most students do not (in my observation). Want to be better? Do more math. –  Jonathan Landrum Feb 7 at 14:06
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@JonathanLandrum There is no shortage of hardworking postdocs who never get tenure. While "X isn't something you are gifted with" is a great motivational mantra (and sometimes work very well for everyman's skills) it might be deadly dangerous for advice on life-long commitments (people do have different predispositions). –  Piotr Migdal Feb 7 at 18:10
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@JonathanLandrum Mathematics (like writing, painting, guitar, basketball, ballet, origami, and duck calling) isn't only something you're gifted with; it's also a practiced skill. And doing it professionally requires not only innate skill and continuous practice and relentless optimism in the face of failure, but lots and lots of luck. –  JeffE Feb 8 at 0:25
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3 Answers 3

Do any mathematicians working in academia have any comments about how one knows if they'll be able to continually generate new ideas to produce publishable research?

I can remember freaking out about this as I was writing my first paper. It was the only publishable work I had ever done, and I remember thinking to myself "What if this is the only idea I have in me? Or what if, instead of getting easier, generating ideas gets even harder as I have to scrounge around in deeper recesses of my brain?"

Fortunately, generating ideas turns out not to be as intimidating as it sounds. In practice, it's very rare to sit down in a chair and say "I shall now think deep thoughts." Instead, any depth comes as a spin-off from much more mundane activities. You read papers, you idly wonder about things, you come up with questions you care about, you figure out how to investigate them, you grapple with technical obstacles, you study things you hadn't realized you needed to know, you chat with colleagues and ask them questions, you work with collaborators, etc. Each of these activities is pretty natural, and they all feed into each other in a complicated web. At any stage you may come up with or run across new ideas, but they are generated organically rather than being something you have to worry about explicitly.

You can expect that a strong graduate program will bring you to the point where you can do this reliably. Of course some people will be faster or more prolific, some will have more striking or creative ideas, some will work on more important questions, etc. You can still improve many of these factors through practice and mentoring, but at that point the question is a little different. Not whether you can do research, but rather how to reach your full potential as a researcher.

So I'd recommend not worrying about this too much. Doing research is a skill that most undergraduates don't have but that graduate schools can teach. Once you get up to speed, generating ideas doesn't end up being a bottleneck. [In fact, you'll end up having more ideas than you have time or energy to investigate yourself. This lets you suggest some to students to help them get started with research, without worrying that you are giving away a limited resource you need for your own research.]

However, given that the opportunity cost of grad school is high and that I already have a stable, good paying job, I would like to know whether I have the talent and ability to be a math professor before I start.

Doing good research is necessary but not sufficient for getting a job in a research university. It's difficult to assess talent and predict career success, but one way to get a crude approximation is by looking at what happened to past students in your position. When you are admitted to graduate school, you can look up former students of advisors you are considering and see what happened to them. For example, you can find lists of students using the Mathematics Genealogy Project, and then you can search for them on the web. This is certainly not perfect: some advisors don't have many students yet, job markets change over time, some former students are just not representative of your situation (if a potential advisor used to be at a less prestigious school, then placement records from that school are not so relevant), etc., and of course there's always random variance. However, it will give you a crude picture. If the advisors you are considering have had many students who got jobs you would like, then maybe you will too. If very few of them got jobs you would find acceptable, then you are taking a much bigger risk.

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+1 for In fact, you'll end up having more ideas than you have time or energy to investigate yourself. This will happen sooner than you think. –  David Ketcheson Feb 7 at 17:54
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I was in the same situation as you.

By good fortune, I lived in a city with an excellent mathematics department. I chose a graduate course that looked interesting and asked the professor if I could sit in on his course. He kindly agreed, and I did so -- including all the homework and a term project. He was impressed, wrote me a rec letter for grad school, and I had the good fortune to succeed -- I am now working as a math professor. So it's possible!

I would definitely advise the same to you if practical.

Also, I recommend that you ask this question to whoever will be writing your rec letters, as they are familiar with what it takes to succeed in mathematics graduate school. (If they are not, then you probably don't want to get letters from them.) If they believe you are strong enough to get accepted to, and succeed in, top programs, then that's a very good sign.

You might consider hedging your bets by only applying to top (say, top 25) graduate programs. You can also get an excellent graduate education at second-tier schools -- indeed, I am a professor at such a school -- but you face longer odds if you graduate from such a school, and if you don't mind the prospect of being admitted nowhere, then being very selective is one way of partially mitigating your long-term risk.

Finally, you might investigate what it is like to work at a not very prestigious institution, such as a regional branch campus of a state university. Would you prefer such a job to your current one? The answer to that question should inform how much of a risk you are willing to take.

Good luck to you!

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Thanks for your response. It's seems like sound advice. However, it would be good to get your thoughts on working as an academic, as opposed to getting a job as an academic. Do you feel you can always competently write new papers? Do you occasionally get anxiety about not being able to solve problems, develop new ideas, write papers, etc? –  user11517 Feb 7 at 15:32
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There is ALWAYS anxiety. See the number of questions on this site alone relating to the imposter syndrome. –  Suresh Feb 7 at 16:14
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One aspect of the question is whether you are willing to move. Even targeting "only" low-mid tier research universities, there are many more good applicants than positions, and if you want to end up in particular area then the risk not to find a job is very high. So, to get higher odds you should be ready to get a PhD somewhere, do a couple post-docs at different places, and then (hopefully) be hired in yet another place.

Concerning your fear with the long-term ability to do math research, I would say that if you manage to find such a job, then you most certainly have what it takes. Sure, some of us loose their way, but it is usually because of particular events and possibly the way they handled it. In all cases I know, at the time of their tenure no one could see a difference between them and ultimately more successful colleagues.

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