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I have been offered a great PhD student position in information theory and machine learning, but in my heart i want to pursue a PhD in pure mathematics, hence I am debating with myself whether or not I should take it. I am looking for advice primarily from mathematicians, and especially those that did not have a straight line path to a career in mathematics.

Some context is necessary. I have a background in engineering, with a B.Sc. in electrical engineering and an M.Sc. in engineering physics (with theoretical physics specialization). This is in Europe, and 3 years bachelor's plus 2 years master's is standard in my country. I graduated a couple of years ago, but I got into engineering quite late, having had an earlier career in music, so I am now in my mid thirties. Since graduation I have worked with signal processing and machine learning in a commercial R&D setting.

Ever since starting my engineering studies I have had a growing interest in mathematics, but it never quite felt practically feasible to pursue full time mathematics studies (recall that I started late; I also had to work part time throughout my studies). However, I did take a few pure mathematics courses like real analysis, abstract algebra, and discrete mathematics while at university. More recently I have started to read some mathematics in my free time (currently Lee's Introduction to Topological Manifolds), but progress is slow since I can only use evenings and weekends.

This spring I started applying to PhD student positions. Primarily in pure mathematics, but also a couple in theoretical aspects of machine learning and physics. I had low hopes for the mathematics positions though, since I just barely fulfilled the formal requirements, and indeed nothing came of those applications. But I did get offered a position at an electrical engineering department where, if I take it, I will be working with information theoretic analysis of machine learning algorithms.

As I see it I have two options:

  1. Take the information theory PhD position. As far as electrical engineering subjects go, I think this is about as mathematical as it gets. I also know from my interviews that I would have a lot of freedom to take courses from the mathematics department as long as they are relevant to the project. Hence I should be able to take courses in measure theoretic probability theory, optimization, perhaps complex and functional analysis, and so on, with the mathematics students. But my research would be constrained to the main subject, and in the end my doctorate would be in electrical engineering.

  2. Start a second master's in pure mathematics, and apply for math PhD positions again next year. The idea is to get more advanced mathematics credits under my belt, and then ideally to get a PhD student position in mathematics before finishing this second master's. But of course it might take the full two years (or longer) before I get anything, if I even get anything at all. With full time studies I believe I would be capable to get perfect or close to perfect grades. I have enough money saved to take a year off from work, but if it takes longer I may have to reduce the studies to part time.

Clearly option 1 is the rational one from a career perspective, but my heart wants to go with option 2. One big worry is my age, since option 2 will take at least one year extra, and, as mentioned earlier, I am already in my mid thirties. I expect that I will want to continue in academia after the PhD, but maybe such a late PhD in mathematics will make that difficult?

I guess my main question is, would it be moronic to go for option 2? I would also love to hear from people who have had similarly strange paths into a career in mathematics, if there are any on the site. What advice can you share?

Clarifications:

  • These are all fully funded positions that I am talking about, as is the norm in my country.

  • There are other good candidates lined up if I don't take the information theory position, so I would not be putting the advisor in a rough spot.

  • I have strong references in machine learning and theoretical physics. In particular I have previous mentors that are happy to write letters of recommendation for me. I think this was an important factor in me getting offered the information theory position.

  • I don't have any family relying on me economically.

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    I guess the number one question is, if you successfully finish a PhD in EE/ML, would you be happy continuing to work in EE/ML or is your current intention to use the PhD programe to transition into pure math? Starting a PhD in a field in which you don't wish to work sounds like a bad idea, unless you have a concrete realistic plan for how you want to leverage that to transition to a different field. On the other hand, starting a PhD in a field which is not your dream field but which is in demand and in which you could realistically see yourself working does not sound like a bad idea at all. May 4 at 22:00
  • @AdamPřenosil Yes, that is certainly a concern with option 1. I worry that I might end up unsatisfied in the long run with being confined to EE/ML, and that I will be stuck there, even if I have a strong mathematical profile in the PhD. It seems an unlikely trajectory to take a PhD in EE/ML and somehow manage to transition into pur mathematics.
    – ummg
    May 4 at 22:11
  • Also, while I think statistics and machine learning is very interesting from a theoretical perspective, I am not one of those people that are super excited about applications of machine learning. If anything, my gut reaction to current developments is often a slight sense of worry... Maybe that will be disheartening in the long run.
    – ummg
    May 4 at 22:15
  • Crossposted: mathoverflow.net/questions/470589/… May 4 at 23:18
  • "One big worry is my age, since option 2 will take at least one year extra" But is one or two years really that much when you're in your mid-30s? If people are prepared to accept someone older than the usual PhD student, does it matter if he's 35 or 37?
    – Stefan
    May 5 at 8:09

5 Answers 5

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First of all, congrats on being accepted to a fully funded PhD program! That's quite an accomplishment. Many people apply for such programs and don't get into any.

Second, "math" and "applied math" are not as siloed as you might think. There are many professors, like me, who maintain a presence in both worlds. It is certainly possible to get a PhD in applied math (or electrical engineering, physics, signal processing, information theory, or machine learning) and still end up doing research that would be classified as "pure math."

To make your decision look carefully at the published papers and research interests of the faculty at the PhD program where you have been accepted. If you say yes, then you'll need to work with one of these people. Are any of them doing work you are passionate about? A PhD is really hard. If you try to get one in a research area you're not passionate about, it's almost impossible (and miserable).

Depending on the specifics, there might be information theory professors whose research consists of probability theory, measure theory, functional analysis, Fourier analysis, etc. There might be machine learning professors whose research consists of probability theory, advanced linear algebra, or topological data analysis (TDA) which includes homological algebra. You can't know unless you look.

You can also inquire about the possibility to take some math courses as part of the PhD program you mentioned. That's usually possible and many applied math professors would be thrilled with a PhD student with a deeper than usual understanding of pure math (because, that can help quite a lot in applied math research). For example, during my PhD in math, I took courses and wrote a thesis to get a master's degree in computer science. I know another person who did the same concurrently getting a PhD in math and a master's in stats. It can even be possible to have a PhD advisor or co-advisor from another department (though, rare). For example, you can imagine working with a machine learning researcher who wants to learn more about TDA, and there's also a math professor who wants to get more into TDA, and they jointly supervise your thesis which you could have done working for either of them separately. My point is: advisors might constrain your research less than you think they will, as long as you can connect it in some way to their broader research program. Another point worth mentioning is that if you go to this program and take a few math courses as you mentioned you'd be allowed to do, then develop a strong relationship with a math professor, you might be able to switch over to the math program, with the blessing of the applied math professors, who in the end will want you to follow your true calling.

That said, all things being equal, a PhD in applied math is better for jobs than a PhD in math, whether you want to be a professor or work in industry. If there's an option to get that PhD even though your research would be heavily pure math, that's the ideal situation for you. You won't know till you check the research of the professors at the place you got accepted.

Lastly, the late PhD should not be held against you. In the US, it would actually be illegal to do so. When you get to that stage, you can try to market your extra experience as a strength.

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  • "all things being equal, a PhD in applied math is better for jobs than a PhD in math, whether you want to be a professor or work in industry". I have seen many people working as math professors, high school math teachers, etc... They may disagree with your statement. May 5 at 0:48
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    @Job_September_2020 My statement is based on actual data. Check the AMS Notices annual reports. I'm a math PhD working as a math professor. I've also been on a dozen hiring committees. The fact is that, like computer science, hiring an applied mathematician is harder than hiring a pure mathematician. We get fewer applicants, and they have more offers from other places. Same for my students going into industry: easier to get a job with applied skills than pure. May 5 at 0:55
  • @DavidWhite Thank you for responding. You raise many good points, though I think the idea about possibly "switching over" to the math program might be too unlikely to let it affect my decision. I will take your advice about looking carefully at the published papers of the faculty at the program to heart. I have already browsed the thesis of the advisor's previous PhD student (and I attended his thesis defence), since my work would closely relate to his. But I will also look at some papers by the advisor himself, and possible others at the department.
    – ummg
    May 5 at 16:24
  • @DavidWhite I have posted a follow-up answer, in case you are interested.
    – ummg
    May 17 at 12:56
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I think the applied math degree is the rational practical solution. But your question says several times that your heart is with the mathematics.

I wouldn't venture advice (I know too little about you) but given the fact that you have no family to support you might entertain the nominally less practical choice. If the math masters disappoints you or it seems as if a pure math phd is not going to happen the year of mathematics can only strengthen a reapplication to an applied math program.

In any case let us know here what you decide and, later, how it worked out.

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  • I hope you are correct that "the year of mathematics can only strengthen a reapplication to an applied math program," and that I would not just appear indecisive.
    – ummg
    May 5 at 16:36
  • I have posted a follow-up answer, in case you are interested.
    – ummg
    May 17 at 12:57
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Follow-up

I ended up not taking the position. The way I reasoned is that I really had two different questions, which I had somewhat conflated:

  1. Do I want to work for ~4 years in a project about information theoretic analysis of machine learning algorithms, at the specific department offering the position, and earn a doctorate in electrical engineering?

  2. If the answer to answer to question 1 is no, then should I start a second master's in mathematics (with the ultimate goal of earning a PhD in mathematics)?

I should therefore evaluate question 1 alone first while trying to ignore question 2. Hence I used the last few days before the decision deadline to study the dissertation of the previous PhD student of the advisor since this is work that I would be extending. I also followed David White's advice to read papers authored by the advisor and other faculty at the department. My conclusion was that these texts did not make me excited enough to warrant embarking on a ~4-year project in the field.

I will likely follow through with the mathematics project of question 2 as well. I feel better about it now that it does not seem like it comes at the expense of the ML/EE doctorate; especially if it turns out not to lead to a doctorate in mathematics in the end.

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  • Great, thanks for the update. If you feel better after having made your decision, it's a strong sign that it was the right choice for you. Good luck. May 17 at 12:58
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Have you applied to any programs where the degree is actually "Applied Mathematics"? This comment may apply to the program you list too though. Often you can still do "pure" research in such a program, proving theorems, etc. You could pursue some theory in probability theory/stochastic processes or statistics and the related.

It sounds like you might have a fixed project you are assigned to though. Without knowing the details of that project, it's hard to say what your options really will be. Do you think you can put up with working on that project for 4+ years? Even if you end up not liking the project, liking the people you work with might make up for some of that. But it might really just depend on your own personality/disposition.

For future employability, the information theory and machine learning is probably the better choice. Of course, it is difficult to predict what the economic and political situation will be when you finish. As long as humanity still exists within a technologically advanced civilisation, what you learn will likely still be quite valuable.

On that note, consider the job prospects of previous graduates. Where do they typically go? Programs often keep a running list of their graduates. Maybe there are existing pipelines to private industry or public sector.

It could also be helpful to look at current job advertisements that sound interesting to you to see their desired applicant qualifications. Take note especially of any computer skills listed. Continue to scan such job adverts every now and then, updating your self-study program on the side if/as needed.

Getting a faculty job in this field will also be more lucrative than math. Your starting salary will be a significant amount above that for a starting math assistant professor. This of course will vary quite a bit depending on your accomplishments and where you apply. But, as mentioned elsewhere, a greater proportion of PhDs in such an applied field will be going to industry, but those academic programs are facing higher and higher demand, so basic economics works in your favor if you want to be faculty in an engineering department or similar.

It's a lot to consider, but it's an important decision. Make sure you can live with your choice, or, at least, if you decide to change course, be honest with yourself about it and take appropriate decisive action.

Good luck!

For what it's worth, I chose applied mathematics and probability theory and am pleased with that choice since the ability to teach statistics is an in demand skill too. I didn't have the computer skills necessary for industry jobs though. That's why I emphasize that latter aspect. I'm tenured faculty in an undergraduate math department. That also brings up another piece to mention: consider the fact that there might be smaller colleges to apply to for faculty positions too. Maybe you will easily qualify for R1, but consider honing your teaching skills too just in case. Not that teaching skills aren't important at an R1, but they will be more important when applying to teaching-focused jobs such as a small liberal arts college (say, one with an engineering or math major).

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You've done work in signal processing and machine learning, you have an experience in possible applications and you are not getting younger. Take the position in applied math, information theory and machine learning. There is a lot of fresh and nontrivial math to deal with and with your experience you are better suited to ask and answer good and relevant questions. You can seek an adviser in this area that has more theoretical inclinations. I say this as a working mathematician who finds the math stemming from computer science very exciting, challenging and fresh.

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